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Your data matches 24 different statistics following compositions of up to 3 maps.
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(click to perform a complete search on your data)
Matching statistic: St000387
(load all 5 compositions to match this statistic)
(load all 5 compositions to match this statistic)
Values
([],1)
=> 0
([],2)
=> 0
([(0,1)],2)
=> 1
([],3)
=> 0
([(1,2)],3)
=> 1
([(0,2),(1,2)],3)
=> 1
([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> 0
([(2,3)],4)
=> 1
([(1,3),(2,3)],4)
=> 1
([(0,3),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2)],4)
=> 2
([(0,3),(1,2),(2,3)],4)
=> 2
([(1,2),(1,3),(2,3)],4)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([],5)
=> 0
([(3,4)],5)
=> 1
([(2,4),(3,4)],5)
=> 1
([(1,4),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> 1
([(1,4),(2,3)],5)
=> 2
([(1,4),(2,3),(3,4)],5)
=> 2
([(0,1),(2,4),(3,4)],5)
=> 2
([(2,3),(2,4),(3,4)],5)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
Description
The matching number of a graph.
For a graph $G$, this is defined as the maximal size of a '''matching''' or '''independent edge set''' (a set of edges without common vertices) contained in $G$.
Matching statistic: St000093
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> 0
([],2)
=> ([],0)
=> 0
([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],0)
=> 0
([(1,2)],3)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],0)
=> 0
([(2,3)],4)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([],5)
=> ([],0)
=> 0
([(3,4)],5)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,4),(2,3)],5)
=> ([],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ? = 3
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,3),(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,7),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,7),(1,2),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(2,7),(3,4),(3,6),(4,5),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,8),(1,2),(1,3),(1,7),(2,3),(2,6),(3,5),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,8),(2,3),(2,4),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(1,2),(1,5),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,7),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,5),(1,6),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,7),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,7),(1,5),(1,6),(2,3),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ? = 3
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,6),(1,4),(2,4),(2,6),(3,5),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,4),(1,6),(1,7),(2,4),(2,5),(2,7),(3,5),(3,6),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 3
([(0,6),(1,2),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,7),(1,2),(1,4),(1,6),(2,4),(2,5),(3,5),(3,6),(3,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,7),(1,5),(1,7),(2,4),(2,6),(2,7),(3,4),(3,5),(3,6),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,7),(6,7)],8)
=> ? = 3
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number $\alpha(G)$ of $G$.
Matching statistic: St000097
(load all 3 compositions to match this statistic)
(load all 3 compositions to match this statistic)
Values
([],1)
=> ([],0)
=> ([],0)
=> 0
([],2)
=> ([],0)
=> ([],0)
=> 0
([(0,1)],2)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],0)
=> ([],0)
=> 0
([(1,2)],3)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
([],4)
=> ([],0)
=> ([],0)
=> 0
([(2,3)],4)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
([],5)
=> ([],0)
=> ([],0)
=> 0
([(3,4)],5)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> 1
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> 2
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ?
=> ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ?
=> ? = 3
([(0,1),(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ?
=> ?
=> ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(0,7),(1,5),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(1,7),(2,6),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7)],8)
=> ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,7),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,7),(1,2),(1,5),(1,6),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,5),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(1,7),(1,8),(2,6),(2,8),(3,6),(3,7),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? = 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(2,7),(3,4),(3,6),(4,5),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,1),(0,2),(0,3),(0,8),(1,2),(1,3),(1,7),(2,3),(2,6),(3,5),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 2
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,8),(2,3),(2,4),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 2
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,3),(0,8),(0,9),(1,2),(1,3),(1,7),(1,9),(2,3),(2,6),(2,9),(3,5),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(1,7),(1,8),(1,9),(2,6),(2,8),(2,9),(3,6),(3,7),(3,9),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(5,9)],10)
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,8),(1,9),(2,3),(2,4),(2,7),(2,9),(3,4),(3,6),(3,9),(4,5),(4,9),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],10)
=> ?
=> ? = 2
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ?
=> ? = 2
([(0,6),(1,6),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,4),(1,5),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,4),(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(5,6)],7)
=> ? = 3
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(1,2),(1,5),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ? = 3
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5)],7)
=> ? = 3
([(0,6),(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,6),(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(1,2),(1,5),(1,7),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,7),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,6),(1,6),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,2),(0,6),(0,8),(1,2),(1,5),(1,7),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 3
([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)
=> ([(0,2),(1,5),(1,6),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(1,4),(1,6),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(5,6)],7)
=> ? = 3
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,3),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,7),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,6),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,6),(1,5),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,2),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,6),(1,2),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(0,7),(1,2),(1,5),(1,8),(2,4),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 3
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,7),(1,5),(1,6),(2,3),(2,4),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,6),(1,8),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,7),(2,6),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8)],9)
=> ? = 2
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)
=> ([(0,4),(0,6),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6)],7)
=> ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ? = 3
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
=> ([(0,6),(0,7),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
Description
The order of the largest clique of the graph.
A clique in a graph $G$ is a subset $U \subseteq V(G)$ such that any pair of vertices in $U$ are adjacent. I.e. the subgraph induced by $U$ is a complete graph.
Matching statistic: St000381
Mp00156: Graphs —line graph⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 75%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000381: Integer compositions ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 75%
Values
([],1)
=> ([],0)
=> ? => ? = 0
([],2)
=> ([],0)
=> ? => ? = 0
([(0,1)],2)
=> ([],1)
=> [1] => 1
([],3)
=> ([],0)
=> ? => ? = 0
([(1,2)],3)
=> ([],1)
=> [1] => 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([],4)
=> ([],0)
=> ? => ? = 0
([(2,3)],4)
=> ([],1)
=> [1] => 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,3),(1,2)],4)
=> ([],2)
=> [2] => 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [2,1] => 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2] => 2
([],5)
=> ([],0)
=> ? => ? = 0
([(3,4)],5)
=> ([],1)
=> [1] => 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([(1,4),(2,3)],5)
=> ([],2)
=> [2] => 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1] => 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [2,2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,1] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? => ? = 2
([],6)
=> ([],0)
=> ? => ? = 0
([(4,5)],6)
=> ([],1)
=> [1] => 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(2,5),(3,4)],6)
=> ([],2)
=> [2] => 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ? => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? => ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? => ? = 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,7),(2,4),(2,6),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,7),(1,2),(1,3),(1,6),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,3),(0,7),(1,2),(1,6),(2,4),(2,6),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,8),(1,2),(1,4),(1,7),(2,3),(2,6),(2,7),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> ? => ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(0,8),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? => ? = 3
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(3,8),(4,5),(4,7),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? => ? = 3
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,7),(1,2),(1,5),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,8),(4,5),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,7),(0,8),(1,4),(1,7),(1,8),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,8),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,7),(0,9),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,6),(3,7),(3,9),(4,5),(4,6),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? => ? = 3
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? => ? = 3
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(3,8),(3,9),(4,5),(4,7),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,6),(0,7),(0,9),(1,4),(1,5),(1,8),(2,3),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ? => ? = 3
Description
The largest part of an integer composition.
Matching statistic: St000382
Mp00156: Graphs —line graph⟶ Graphs
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 75%
Mp00324: Graphs —chromatic difference sequence⟶ Integer compositions
St000382: Integer compositions ⟶ ℤResult quality: 22% ●values known / values provided: 22%●distinct values known / distinct values provided: 75%
Values
([],1)
=> ([],0)
=> ? => ? = 0
([],2)
=> ([],0)
=> ? => ? = 0
([(0,1)],2)
=> ([],1)
=> [1] => 1
([],3)
=> ([],0)
=> ? => ? = 0
([(1,2)],3)
=> ([],1)
=> [1] => 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([],4)
=> ([],0)
=> ? => ? = 0
([(2,3)],4)
=> ([],1)
=> [1] => 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> [1,1] => 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,3),(1,2)],4)
=> ([],2)
=> [2] => 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> [2,1] => 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2] => 2
([],5)
=> ([],0)
=> ? => ? = 0
([(3,4)],5)
=> ([],1)
=> [1] => 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> [1,1] => 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([(1,4),(2,3)],5)
=> ([],2)
=> [2] => 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> [2,1] => 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,1,1,1] => 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> [2,2] => 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> [2,2,2] => 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> [2,2,2,1] => 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> [2,2] => 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> [2,1,1] => 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> [2,2,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> [2,2,1,1] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> [2,2,1] => 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> [2,2,2] => 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> [2,2,2] => 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> [2,2,2] => 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> [2,2,2,1] => 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> [2,2,2,1] => 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? => ? = 2
([],6)
=> ([],0)
=> ? => ? = 0
([(4,5)],6)
=> ([],1)
=> [1] => 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> [1,1] => 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> [1,1,1] => 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> [1,1,1,1] => 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> [1,1,1,1,1] => 1
([(2,5),(3,4)],6)
=> ([],2)
=> [2] => 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> [2,1] => 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ? => ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? => ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? => ? = 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,7),(2,4),(2,6),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,4),(0,7),(1,2),(1,3),(1,6),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,1),(0,3),(0,7),(1,2),(1,6),(2,4),(2,6),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,8),(1,2),(1,4),(1,7),(2,3),(2,6),(2,7),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,7),(6,7)],8)
=> ? => ? = 3
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6),(6,7)],8)
=> ? => ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,8),(3,4),(3,6),(4,7),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,2),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,6),(0,7),(1,4),(1,5),(2,3),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,2),(1,3),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(0,8),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? => ? = 3
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(3,4),(3,5),(3,6),(3,8),(4,5),(4,7),(4,8),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,6),(0,7),(0,8),(0,9),(1,2),(1,4),(1,5),(1,7),(1,9),(2,3),(2,5),(2,7),(2,8),(3,4),(3,5),(3,6),(3,8),(4,5),(4,6),(4,9),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? => ? = 3
([(0,4),(0,5),(1,2),(1,4),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,7),(1,2),(1,5),(1,7),(1,8),(2,3),(2,6),(2,8),(3,6),(3,8),(4,5),(4,6),(4,7),(5,6),(5,7),(5,8),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,7),(0,8),(1,4),(1,7),(1,8),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,8),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(0,7),(0,9),(1,2),(1,4),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,6),(3,7),(3,9),(4,5),(4,6),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? => ? = 3
([(0,3),(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ? => ? = 3
([(0,1),(0,2),(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,7),(0,8),(1,2),(1,5),(1,6),(2,3),(2,4),(3,4),(3,6),(3,8),(3,9),(4,5),(4,7),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? => ? = 3
([(0,4),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,6),(0,7),(0,9),(1,4),(1,5),(1,8),(2,3),(2,5),(2,7),(2,8),(2,9),(3,4),(3,6),(3,8),(3,9),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,9),(7,9),(8,9)],10)
=> ? => ? = 3
Description
The first part of an integer composition.
Matching statistic: St001337
Values
([],1)
=> ([],1)
=> ([],0)
=> ? = 0
([],2)
=> ([],2)
=> ([],0)
=> ? = 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],0)
=> ? = 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],4)
=> ([],0)
=> ? = 0
([(2,3)],4)
=> ([(2,3)],4)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([],5)
=> ([],5)
=> ([],0)
=> ? = 0
([(3,4)],5)
=> ([(3,4)],5)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([],6)
=> ([],6)
=> ([],0)
=> ? = 0
([(4,5)],6)
=> ([(4,5)],6)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([],2)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
Description
The upper domination number of a graph.
This is the maximum cardinality of a minimal dominating set of $G$.
The smallest graph with different upper irredundance number and upper domination number has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
Values
([],1)
=> ([],1)
=> ([],0)
=> ? = 0
([],2)
=> ([],2)
=> ([],0)
=> ? = 0
([(0,1)],2)
=> ([(0,1)],2)
=> ([],1)
=> 1
([],3)
=> ([],3)
=> ([],0)
=> ? = 0
([(1,2)],3)
=> ([(1,2)],3)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([],4)
=> ([],4)
=> ([],0)
=> ? = 0
([(2,3)],4)
=> ([(2,3)],4)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2)],4)
=> ([(0,3),(1,2)],4)
=> ([],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([],5)
=> ([],5)
=> ([],0)
=> ? = 0
([(3,4)],5)
=> ([(3,4)],5)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(1,4),(2,3)],5)
=> ([(1,4),(2,3)],5)
=> ([],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([],6)
=> ([],6)
=> ([],0)
=> ? = 0
([(4,5)],6)
=> ([(4,5)],6)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 1
([(2,5),(3,4)],6)
=> ([(2,5),(3,4)],6)
=> ([],2)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> 2
([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 2
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ? = 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ?
=> ? = 3
Description
The upper irredundance number of a graph.
A set $S$ of vertices is irredundant, if there is no vertex in $S$, whose closed neighbourhood is contained in the union of the closed neighbourhoods of the other vertices of $S$.
The upper irredundance number is the largest size of a maximal irredundant set.
The smallest graph with different upper irredundance number and upper domination number [[St001337]] has eight vertices. It is obtained from the disjoint union of two copies of $K_4$ by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St001674
Values
([],1)
=> ([],0)
=> ([],1)
=> 1 = 0 + 1
([],2)
=> ([],0)
=> ([],1)
=> 1 = 0 + 1
([(0,1)],2)
=> ([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
([],3)
=> ([],0)
=> ([],1)
=> 1 = 0 + 1
([(1,2)],3)
=> ([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([],4)
=> ([],0)
=> ([],1)
=> 1 = 0 + 1
([(2,3)],4)
=> ([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
([],5)
=> ([],0)
=> ([],1)
=> 1 = 0 + 1
([(3,4)],5)
=> ([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,2),(1,2)],3)
=> 3 = 2 + 1
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> 3 = 2 + 1
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,4),(0,5),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 3 = 2 + 1
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> 3 = 2 + 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(0,7),(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> 3 = 2 + 1
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,4),(0,5),(0,6),(1,2),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)
=> 3 = 2 + 1
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,2),(0,3),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,8),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(0,9),(1,2),(1,3),(1,7),(1,8),(1,9),(2,3),(2,5),(2,6),(2,8),(2,9),(3,4),(3,6),(3,7),(3,9),(4,5),(4,6),(4,7),(4,9),(5,6),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2 + 1
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(0,10),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,8),(2,9),(2,10),(3,4),(3,5),(3,6),(3,7),(3,10),(4,5),(4,7),(4,9),(4,10),(5,6),(5,8),(5,10),(6,7),(6,8),(6,10),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 2 + 1
([],6)
=> ([],0)
=> ([],1)
=> 1 = 0 + 1
([(4,5)],6)
=> ([],1)
=> ([(0,1)],2)
=> 2 = 1 + 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([(0,1),(0,2),(1,2)],3)
=> 2 = 1 + 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> 2 = 1 + 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> 2 = 1 + 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 2 = 1 + 1
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,6),(0,7),(0,8),(1,2),(1,5),(1,7),(1,8),(2,4),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ? = 2 + 1
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,5),(2,7),(2,8),(2,9),(3,4),(3,7),(3,8),(3,9),(4,5),(4,6),(4,8),(4,9),(5,6),(5,8),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,5),(0,7),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(0,7),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,6),(0,7),(1,2),(1,4),(1,5),(1,7),(2,3),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,2),(0,5),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,6),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,3),(0,4),(0,7),(1,2),(1,6),(1,7),(2,5),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,7),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(1,2),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,8),(1,2),(1,4),(1,7),(1,8),(2,3),(2,7),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(0,7),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,1),(0,2),(0,6),(0,7),(0,8),(1,2),(1,5),(1,7),(1,8),(2,5),(2,6),(2,8),(3,4),(3,5),(3,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ([(0,3),(0,6),(0,7),(0,8),(1,2),(1,5),(1,6),(1,8),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,8),(1,2),(1,4),(1,6),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,8),(2,3),(2,4),(2,5),(2,7),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,8),(1,2),(1,4),(1,6),(1,7),(1,8),(2,3),(2,6),(2,7),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,5),(0,6),(0,7),(0,9),(1,2),(1,4),(1,6),(1,8),(1,9),(2,3),(2,6),(2,8),(2,9),(3,4),(3,5),(3,7),(3,8),(3,9),(4,5),(4,7),(4,8),(4,9),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,8),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,6),(2,7),(2,8),(3,4),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,7),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(2,3),(2,4),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,3),(0,6),(0,7),(0,8),(0,9),(1,4),(1,5),(1,6),(1,7),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,6),(3,7),(3,8),(3,9),(4,5),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ([(0,2),(0,3),(0,7),(0,8),(0,9),(1,2),(1,3),(1,5),(1,6),(1,9),(2,5),(2,6),(2,9),(3,7),(3,8),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3 + 1
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,5),(0,6),(0,7),(0,8),(0,10),(1,3),(1,4),(1,7),(1,8),(1,9),(1,10),(2,3),(2,4),(2,5),(2,6),(2,9),(2,10),(3,5),(3,6),(3,9),(3,10),(4,7),(4,8),(4,9),(4,10),(5,6),(5,8),(5,9),(5,10),(6,7),(6,9),(6,10),(7,8),(7,9),(7,10),(8,9),(8,10),(9,10)],11)
=> ? = 3 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,1),(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,4),(1,6),(1,8),(2,3),(2,5),(2,8),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,2),(0,3),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,8),(2,5),(2,7),(2,8),(3,4),(3,6),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 2 + 1
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,2),(0,3),(0,7),(0,8),(0,9),(1,2),(1,3),(1,5),(1,6),(1,9),(2,6),(2,8),(2,9),(3,5),(3,7),(3,9),(4,5),(4,6),(4,7),(4,8),(4,9),(5,6),(5,7),(5,8),(5,9),(6,7),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
=> ? = 3 + 1
([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(0,7),(1,2),(1,3),(1,7),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,7),(2,4),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,6),(3,5),(3,6),(4,6),(5,6)],7)
=> ([(0,1),(0,5),(0,7),(1,4),(1,7),(2,3),(2,4),(2,6),(2,7),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,5),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,6),(1,2),(1,3),(1,5),(2,5),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ([(0,4),(0,6),(0,7),(1,2),(1,3),(1,5),(1,7),(2,5),(2,6),(2,7),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ? = 3 + 1
([(0,1),(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(1,3),(1,4),(1,7),(2,4),(2,6),(3,5),(3,6),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,2),(0,5),(0,8),(1,3),(1,4),(1,7),(1,8),(2,4),(2,6),(2,8),(3,5),(3,6),(3,7),(3,8),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 3 + 1
Description
The number of vertices of the largest induced star graph in the graph.
Matching statistic: St001316
Values
([],1)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([],2)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([],4)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> 2
([],5)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],1)
=> 1
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ?
=> ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ?
=> ? = 2
([],6)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],1)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],1)
=> 1
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> ?
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ?
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ([(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7)],9)
=> ?
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> ?
=> ? = 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,5),(3,7),(4,5),(4,6),(4,7),(6,7)],8)
=> ?
=> ? = 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,4),(1,5),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(1,4),(2,5),(2,7),(3,5),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(1,6),(2,3),(2,4),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,5),(2,3),(2,6),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,2),(1,5),(1,6),(2,3),(2,4),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,6),(1,7),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ?
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,5),(1,4),(1,7),(2,3),(2,7),(3,6),(4,6),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ([(0,3),(0,7),(0,8),(1,2),(1,5),(1,6),(2,7),(2,8),(3,5),(3,6),(4,5),(4,6),(4,7),(4,8),(5,8),(6,7)],9)
=> ?
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ?
=> ? = 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,8),(6,7)],9)
=> ?
=> ? = 3
Description
The domatic number of a graph.
This is the maximal size of a partition of the vertices into dominating sets.
Matching statistic: St001656
Values
([],1)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([],2)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([(0,1)],2)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([],3)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([(1,2)],3)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
([(0,1),(0,2),(1,2)],3)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([],4)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([(2,3)],4)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
([(0,3),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(0,3),(1,2)],4)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(2,3)],4)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(2,3)],4)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> 2
([],5)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([(3,4)],5)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
([(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],1)
=> 1
([(1,4),(2,3)],5)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(3,4)],5)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(0,1),(2,4),(3,4)],5)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,4),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(2,4),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,2),(0,3),(1,2),(1,3)],4)
=> ([(0,3),(1,2)],4)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1)],2)
=> 2
([(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(1,4),(2,3)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(1,5),(2,4),(3,4),(3,5)],6)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ?
=> ? = 2
([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,1),(2,3),(2,4),(3,4)],5)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(1,4),(2,3),(3,4)],5)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,1)],2)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,4),(2,3)],5)
=> 2
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
=> ([(0,1)],2)
=> 2
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(1,4),(2,3)],6)
=> ([(0,1)],2)
=> 2
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 2
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,1),(0,4),(0,5),(0,7),(0,8),(1,2),(1,3),(1,7),(1,8),(2,3),(2,5),(2,6),(2,8),(3,4),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,3),(2,4),(3,6),(3,8),(4,5),(4,7),(5,8),(6,7)],9)
=> ?
=> ? = 2
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(4,5),(4,7),(4,9),(5,6),(5,8),(6,7),(6,8),(7,9),(8,9)],10)
=> ([(0,7),(0,8),(0,9),(1,4),(1,6),(1,9),(2,3),(2,6),(2,8),(3,5),(3,9),(4,5),(4,8),(5,7),(6,7)],10)
=> ?
=> ? = 2
([],6)
=> ([],0)
=> ([],0)
=> ([],0)
=> ? = 0
([(4,5)],6)
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
([(3,5),(4,5)],6)
=> ([(0,1)],2)
=> ([],2)
=> ([],1)
=> 1
([(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4)
=> ([],4)
=> ([],1)
=> 1
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
=> ([],5)
=> ([],1)
=> 1
([(2,5),(3,4)],6)
=> ([],2)
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 2
([(2,5),(3,4),(4,5)],6)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> ([(0,1)],2)
=> 2
([(1,2),(3,5),(4,5)],6)
=> ([(1,2)],3)
=> ([(0,2),(1,2)],3)
=> ([(0,1)],2)
=> 2
([(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(1,2)],3)
=> ([],3)
=> ([],1)
=> 1
([(1,5),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(1,2),(1,3),(2,3)],4)
=> ([(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,1),(2,5),(3,5),(4,5)],6)
=> ([(1,2),(1,3),(2,3)],4)
=> ([(0,3),(1,3),(2,3)],4)
=> ([(0,1)],2)
=> 2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7)
=> ?
=> ? = 2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)
=> ?
=> ? = 2
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7)
=> ?
=> ? = 2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ?
=> ? = 2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,6),(1,7),(1,8),(2,3),(2,5),(2,7),(2,8),(3,4),(3,7),(3,8),(4,5),(4,6),(4,8),(5,6),(5,8),(6,8),(7,8)],9)
=> ([(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7)],9)
=> ?
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,5),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7)
=> ?
=> ? = 3
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,2),(0,5),(0,7),(1,2),(1,4),(1,6),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(2,6),(3,5),(3,7),(4,5),(4,6),(4,7),(6,7)],8)
=> ?
=> ? = 3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(1,2),(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,4),(0,5),(1,4),(1,5),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,6),(2,3),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,7),(2,3),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(1,7),(2,5),(2,7),(3,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,5),(0,6),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
=> ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 2
([(0,5),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,1),(0,2),(0,6),(0,7),(1,2),(1,5),(1,7),(2,5),(2,6),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)
=> ?
=> ? = 2
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5)],7)
=> ([(0,1),(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(1,2),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ([(0,6),(0,7),(1,2),(1,3),(1,4),(2,5),(2,7),(3,5),(3,6),(4,6),(4,7),(5,6),(5,7)],8)
=> ?
=> ? = 3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,5),(1,6),(2,3),(2,4),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(2,7),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,5),(2,3),(2,6),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,1),(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(1,7),(2,3),(2,6),(2,7),(3,4),(3,5),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ([(0,7),(1,2),(1,5),(1,6),(2,3),(2,4),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(1,8),(2,3),(2,6),(2,8),(3,4),(3,5),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,6),(1,7),(2,3),(2,4),(2,5),(3,7),(3,8),(4,6),(4,8),(5,6),(5,7),(6,8),(7,8)],9)
=> ?
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,3),(0,4),(0,5),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(3,4),(3,7),(4,6),(5,6),(5,7),(6,7)],8)
=> ([(0,5),(0,7),(1,4),(1,7),(2,3),(2,4),(2,5),(3,6),(3,7),(4,6),(5,6),(6,7)],8)
=> ?
=> ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,5),(1,6),(1,7),(2,3),(2,4),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,7),(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,7),(6,7)],8)
=> ?
=> ? = 2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,3),(1,4),(1,7),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,5),(1,4),(1,7),(2,3),(2,7),(3,6),(4,6),(5,6),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,3),(0,6),(0,7),(0,8),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,5),(2,8),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,6),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,8),(1,5),(1,7),(2,4),(2,7),(3,4),(3,5),(3,8),(4,6),(5,6),(6,7),(6,8),(7,8)],9)
=> ?
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(4,7),(4,8),(5,6),(5,8),(6,7),(7,8)],9)
=> ([(0,3),(0,7),(0,8),(1,2),(1,5),(1,6),(2,7),(2,8),(3,5),(3,6),(4,5),(4,6),(4,7),(4,8),(5,8),(6,7)],9)
=> ?
=> ? = 3
([(0,4),(0,5),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(0,8),(1,3),(1,4),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,9),(3,5),(3,6),(3,9),(4,7),(4,8),(4,9),(5,6),(5,8),(5,9),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
=> ([(0,9),(1,4),(1,7),(1,8),(2,3),(2,5),(2,6),(3,7),(3,8),(4,5),(4,6),(5,8),(5,9),(6,7),(6,9),(7,9),(8,9)],10)
=> ?
=> ? = 3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(1,2),(1,4),(1,6),(2,3),(2,6),(3,4),(3,5),(4,5),(5,6)],7)
=> ([(0,4),(0,5),(1,2),(1,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
=> ?
=> ? = 3
([(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7)
=> ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
=> ?
=> ? = 2
([(0,5),(1,3),(1,4),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,5),(0,6),(0,7),(1,2),(1,4),(1,6),(2,3),(2,5),(3,4),(3,5),(3,7),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(1,4),(1,6),(2,3),(2,5),(3,6),(3,7),(4,5),(4,7),(5,7),(6,7)],8)
=> ?
=> ? = 3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)
=> ([(0,6),(0,7),(1,4),(1,5),(2,5),(2,7),(3,4),(3,6),(4,7),(5,6)],8)
=> ?
=> ? = 2
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)
=> ([(0,2),(0,3),(0,7),(0,8),(1,2),(1,3),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ([(0,7),(0,8),(1,5),(1,6),(2,6),(2,8),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,8),(6,7)],9)
=> ?
=> ? = 3
Description
The monophonic position number of a graph.
A subset $M$ of the vertex set of a graph is a monophonic position set if no three vertices of $M$ lie on a common induced path. The monophonic position number is the size of a largest monophonic position set.
The following 14 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St000741The Colin de Verdière graph invariant. St000514The number of invariant simple graphs when acting with a permutation of given cycle type. St000515The number of invariant set partitions when acting with a permutation of given cycle type. St000284The Plancherel distribution on integer partitions. St000698The number of 2-rim hooks removed from an integer partition to obtain its associated 2-core. St000704The number of semistandard tableaux on a given integer partition with minimal maximal entry. St000901The cube of the number of standard Young tableaux with shape given by the partition. St001128The exponens consonantiae of a partition. St000936The number of even values of the symmetric group character corresponding to the partition. St000938The number of zeros of the symmetric group character corresponding to the partition. St000940The number of characters of the symmetric group whose value on the partition is zero. St000941The number of characters of the symmetric group whose value on the partition is even. St001124The multiplicity of the standard representation in the Kronecker square corresponding to a partition.
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