- FindStat

Your data matches 3 different statistics following compositions of up to 3 maps.
(click to perform a complete search on your data)

Matching statistic: St000786
(load all 3 compositions to match this statistic)

Mapping

Mp00156: Graphs —line graph⟶ Graphs
Mp00203: Graphs —cone⟶ Graphs
St000786: Graphs ⟶ ℤResult quality: 100% ●values known / values provided: 100%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],0)
 => ([],1)
 => 1
([],2)
 => ([],0)
 => ([],1)
 => 1
([(0,1)],2)
 => ([],1)
 => ([(0,1)],2)
 => 1
([],3)
 => ([],0)
 => ([],1)
 => 1
([(1,2)],3)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(0,2),(1,2)],3)
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 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)
 => 1
([],4)
 => ([],0)
 => ([],1)
 => 1
([(2,3)],4)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(1,3),(2,3)],4)
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 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)
 => 1
([(0,3),(1,2)],4)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 2
([(0,3),(1,2),(2,3)],4)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(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)
 => 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)
 => 2
([(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)
 => 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)
 => ([(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)
 => 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)
 => ([(0,2),(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)
 => 2
([],5)
 => ([],0)
 => ([],1)
 => 1
([(3,4)],5)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(2,4),(3,4)],5)
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 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)
 => 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)
 => 1
([(1,4),(2,3)],5)
 => ([],2)
 => ([(0,2),(1,2)],3)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(0,2),(1,2)],3)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,1),(2,4),(3,4)],5)
 => ([(1,2)],3)
 => ([(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(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)
 => 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)
 => 2
([(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)
 => 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)
 => ([(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
([(1,3),(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)
 => 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)
 => ([(0,1),(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)
 => ([(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)
 => 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)
 => ([(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)
 => 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)
 => ([(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)
 => 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)
 => ([(0,3),(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)
 => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(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)
 => 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),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => 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)
 => ([(0,4),(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)
 => 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),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => 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)
 => ([(0,3),(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)
 => 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)
 => ([(0,4),(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)
 => 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)
 => ([(0,2),(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)
 => 2
([],6)
 => ([],0)
 => ([],1)
 => 1
([(4,5)],6)
 => ([],1)
 => ([(0,1)],2)
 => 1
([(3,5),(4,5)],6)
 => ([(0,1)],2)
 => ([(0,1),(0,2),(1,2)],3)
 => 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)
 => 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)
 => 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)
 => 1

Description

The maximal number of occurrences of a colour in a proper colouring of a graph. To any proper colouring with the minimal number of colours possible we associate the integer partition recording how often each colour is used. This statistic records the largest part occurring in any of these partitions. For example, the graph on six vertices consisting of a square together with two attached triangles - ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) in the list of values - is three-colourable and admits two colouring schemes,

$[2,2,2]$ and

$[3,2,1]$ . Therefore, the statistic on this graph is

$3$ .

Matching statistic: St001337
(load all 2 compositions to match this statistic)

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001337: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => ([],0)
 => ? = 1
([],2)
 => ([],2)
 => ([],2)
 => ([],0)
 => ? = 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => ([],3)
 => ([],0)
 => ? = 1
([(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([],1)
 => 1
([(0,2),(1,2)],3)
 => ([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => ([],4)
 => ([],4)
 => ([],0)
 => ? = 1
([(2,3)],4)
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ([],1)
 => 1
([(1,3),(2,3)],4)
 => ([(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)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(1,2)],4)
 => ([(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)
 => ([(0,3),(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)
 => ([(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)
 => ([(0,3),(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)
 => ([(0,1),(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)
 => ([(0,1),(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)
 => ([(0,1),(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)
 => ([],5)
 => ([],0)
 => ? = 1
([(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ([],1)
 => 1
([(2,4),(3,4)],5)
 => ([(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)
 => ([(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)
 => ([(0,4),(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)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(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)
 => ([(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)
 => ([(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)
 => ([(0,4),(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
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(0,4),(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)
 => ([(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)
 => ([(0,4),(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)
 => ([(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)
 => ([(0,4),(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)
 => ([(0,4),(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),(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)
 => ([(0,1),(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
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(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),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(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)
 => ([(0,3),(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)
 => ([(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)
 => ([(0,1),(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),(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)
 => ([(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
([],6)
 => ([],6)
 => ([],6)
 => ([],0)
 => ? = 1
([(4,5)],6)
 => ([(4,5)],6)
 => ([(4,5)],6)
 => ([],1)
 => 1
([(3,5),(4,5)],6)
 => ([(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)
 => ([(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)
 => ([(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)
 => ([(0,5),(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)
 => ([(2,5),(3,4)],6)
 => ([],2)
 => 2
([(2,5),(3,4),(4,5)],6)
 => ([(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),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(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)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,7),(0,8),(0,9),(1,2),(1,3),(1,5),(1,6),(1,9),(2,3),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(0,6),(1,4),(1,8),(1,9),(2,3),(2,4),(2,6),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(2,4),(2,5),(2,7),(3,5),(3,6),(3,9),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ? = 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)
 => ([(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),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(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)
 => ? = 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)
 => ([(0,5),(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),(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),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,6),(1,8),(1,9),(2,5),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(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
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,6),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
([(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)
 => ([(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),(1,4),(2,3),(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),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,8),(1,3),(1,7),(2,3),(2,4),(2,5),(2,6),(3,5),(3,7),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(2,7),(2,8),(3,4),(3,5),(3,7),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
 => ? = 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,8),(1,2),(1,3),(1,4),(1,7),(2,3),(2,4),(2,6),(3,6),(3,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(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),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(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)
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,7),(4,8),(5,7),(6,8),(7,8)],9)
 => ? = 3
([],7)
 => ([],7)
 => ([],7)
 => ([],0)
 => ? = 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(0,8),(0,9),(1,2),(1,3),(1,5),(1,6),(1,9),(2,3),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,2),(1,3),(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)
 => ?
 => ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(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)
 => ? = 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,5),(0,6),(1,4),(1,8),(1,9),(2,3),(2,4),(2,6),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(2,4),(2,5),(2,7),(3,5),(3,6),(3,9),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(1,2),(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)
 => ?
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(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),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,1),(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)
 => ?
 => ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(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
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,6),(1,8),(1,9),(2,5),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,6),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,8),(1,3),(1,7),(2,3),(2,4),(2,5),(2,6),(3,5),(3,7),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 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
(load all 2 compositions to match this statistic)

Mapping

Mp00117: Graphs —Ore closure⟶ Graphs
Mp00157: Graphs —connected complement⟶ Graphs
Mp00156: Graphs —line graph⟶ Graphs
St001338: Graphs ⟶ ℤResult quality: 66% ●values known / values provided: 66%●distinct values known / distinct values provided: 100%

Values

([],1)
 => ([],1)
 => ([],1)
 => ([],0)
 => ? = 1
([],2)
 => ([],2)
 => ([],2)
 => ([],0)
 => ? = 1
([(0,1)],2)
 => ([(0,1)],2)
 => ([(0,1)],2)
 => ([],1)
 => 1
([],3)
 => ([],3)
 => ([],3)
 => ([],0)
 => ? = 1
([(1,2)],3)
 => ([(1,2)],3)
 => ([(1,2)],3)
 => ([],1)
 => 1
([(0,2),(1,2)],3)
 => ([(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)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([],4)
 => ([],4)
 => ([],4)
 => ([],0)
 => ? = 1
([(2,3)],4)
 => ([(2,3)],4)
 => ([(2,3)],4)
 => ([],1)
 => 1
([(1,3),(2,3)],4)
 => ([(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)
 => ([(0,3),(1,3),(2,3)],4)
 => ([(0,1),(0,2),(1,2)],3)
 => 1
([(0,3),(1,2)],4)
 => ([(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)
 => ([(0,3),(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)
 => ([(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)
 => ([(0,3),(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)
 => ([(0,1),(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)
 => ([(0,1),(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)
 => ([(0,1),(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)
 => ([],5)
 => ([],0)
 => ? = 1
([(3,4)],5)
 => ([(3,4)],5)
 => ([(3,4)],5)
 => ([],1)
 => 1
([(2,4),(3,4)],5)
 => ([(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)
 => ([(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)
 => ([(0,4),(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)
 => ([(1,4),(2,3)],5)
 => ([],2)
 => 2
([(1,4),(2,3),(3,4)],5)
 => ([(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)
 => ([(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)
 => ([(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)
 => ([(0,4),(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
([(1,4),(2,3),(2,4),(3,4)],5)
 => ([(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)
 => ([(0,4),(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)
 => ([(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)
 => ([(0,4),(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)
 => ([(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)
 => ([(0,4),(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)
 => ([(0,4),(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),(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)
 => ([(0,1),(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
([(0,1),(2,3),(2,4),(3,4)],5)
 => ([(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),(2,4),(3,4)],5)
 => ([(0,1),(0,4),(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)
 => ([(0,3),(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)
 => ([(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)
 => ([(0,1),(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),(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)
 => ([(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
([],6)
 => ([],6)
 => ([],6)
 => ([],0)
 => ? = 1
([(4,5)],6)
 => ([(4,5)],6)
 => ([(4,5)],6)
 => ([],1)
 => 1
([(3,5),(4,5)],6)
 => ([(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)
 => ([(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)
 => ([(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)
 => ([(0,5),(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)
 => ([(2,5),(3,4)],6)
 => ([],2)
 => 2
([(2,5),(3,4),(4,5)],6)
 => ([(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),(3,5),(4,5)],6)
 => ([(1,2)],3)
 => 2
([(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => 2
([(0,1),(2,5),(3,5),(4,5)],6)
 => ([(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)
 => ([(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4)
 => 2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,5),(3,4),(4,5)],6)
 => ([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,7),(0,8),(0,9),(1,2),(1,3),(1,5),(1,6),(1,9),(2,3),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(1,5),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)
 => 2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,3),(3,4),(4,5)],6)
 => ([(0,2),(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,1),(0,5),(0,6),(1,4),(1,8),(1,9),(2,3),(2,4),(2,6),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(2,4),(2,5),(2,7),(3,5),(3,6),(3,9),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ? = 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)
 => ([(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),(2,4),(2,5),(3,4),(3,5)],6)
 => ([(0,5),(1,4),(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)
 => ? = 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)
 => ([(0,5),(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),(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),(2,3),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,6),(1,8),(1,9),(2,5),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(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
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)
 => ([(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,6),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
([(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)
 => ([(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),(1,4),(2,3),(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),(1,5),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,1),(0,4),(0,8),(1,3),(1,7),(2,3),(2,4),(2,5),(2,6),(3,5),(3,7),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)
 => ([(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)
 => ? = 2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,5),(1,4),(2,3),(2,4),(3,5)],6)
 => ([(0,1),(0,3),(0,5),(1,2),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,2),(0,5),(0,6),(0,8),(1,2),(1,3),(1,4),(1,7),(2,7),(2,8),(3,4),(3,5),(3,7),(3,9),(4,6),(4,8),(4,9),(5,6),(5,7),(5,9),(6,8),(6,9),(7,9),(8,9)],10)
 => ? = 3
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(0,5),(0,8),(1,2),(1,3),(1,4),(1,7),(2,3),(2,4),(2,6),(3,6),(3,7),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)
 => ([(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),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)
 => ([(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),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)
 => ([(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)
 => ? = 2
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)
 => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6)
 => ([(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,7),(4,8),(5,7),(6,8),(7,8)],9)
 => ? = 3
([],7)
 => ([],7)
 => ([],7)
 => ([],0)
 => ? = 1
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,7),(0,8),(0,9),(1,2),(1,3),(1,5),(1,6),(1,9),(2,3),(2,4),(2,6),(2,8),(3,4),(3,5),(3,7),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)
 => ([(0,6),(1,2),(1,3),(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)
 => ?
 => ? = 2
([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(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)
 => ? = 2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(1,3),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,1),(0,5),(0,6),(1,4),(1,8),(1,9),(2,3),(2,4),(2,6),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,8),(4,9),(5,6),(5,7),(5,8),(6,7),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,7),(0,8),(0,9),(1,2),(1,4),(1,6),(1,8),(2,4),(2,5),(2,7),(3,5),(3,6),(3,9),(4,5),(4,6),(4,7),(4,8),(5,6),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)
 => ([(0,2),(0,6),(1,3),(1,4),(1,5),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 2
([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)
 => ([(0,5),(0,6),(1,2),(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)
 => ?
 => ? = 2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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)
 => ? = 2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(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),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(5,7),(6,7)],8)
 => ? = 2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)
 => ([(0,1),(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)
 => ?
 => ? = 2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(2,6),(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
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,6),(1,8),(1,9),(2,5),(2,7),(2,9),(3,4),(3,5),(3,7),(3,8),(4,6),(4,7),(4,8),(5,7),(5,9),(6,8),(6,9),(7,8),(7,9),(8,9)],10)
 => ? = 3
([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(0,5),(0,6),(1,2),(1,3),(1,4),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)
 => ([(1,4),(1,6),(2,3),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(0,3),(0,6),(0,7),(1,2),(1,4),(1,5),(2,4),(2,6),(2,8),(3,5),(3,7),(3,8),(4,5),(4,6),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 3
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)
 => ([(0,2),(0,3),(0,6),(1,3),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)
 => ?
 => ? = 3
([(2,3),(2,6),(3,5),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)
 => ([(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
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)
 => ([(1,4),(1,5),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)
 => ([(0,1),(0,4),(0,8),(1,3),(1,7),(2,3),(2,4),(2,5),(2,6),(3,5),(3,7),(4,6),(4,8),(5,6),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
 => ? = 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].