Values
=>
Cc0020;cc-rep
([],0)=>0
([],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)=>2
([],4)=>0
([(2,3)],4)=>1
([(1,3),(2,3)],4)=>1
([(0,3),(1,3),(2,3)],4)=>2
([(0,3),(1,2)],4)=>1
([(0,3),(1,2),(2,3)],4)=>1
([(1,2),(1,3),(2,3)],4)=>2
([(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)=>3
([],5)=>0
([(3,4)],5)=>1
([(2,4),(3,4)],5)=>1
([(1,4),(2,4),(3,4)],5)=>2
([(0,4),(1,4),(2,4),(3,4)],5)=>2
([(1,4),(2,3)],5)=>1
([(1,4),(2,3),(3,4)],5)=>1
([(0,1),(2,4),(3,4)],5)=>1
([(2,3),(2,4),(3,4)],5)=>2
([(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)=>3
([(0,4),(1,3),(2,3),(2,4)],5)=>1
([(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
([(0,4),(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)=>3
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5)=>3
([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>3
([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5)=>4
([],6)=>0
([(4,5)],6)=>1
([(3,5),(4,5)],6)=>1
([(2,5),(3,5),(4,5)],6)=>2
([(1,5),(2,5),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,5),(3,5),(4,5)],6)=>2
([(2,5),(3,4)],6)=>1
([(2,5),(3,4),(4,5)],6)=>1
([(1,2),(3,5),(4,5)],6)=>1
([(3,4),(3,5),(4,5)],6)=>2
([(1,5),(2,5),(3,4),(4,5)],6)=>2
([(0,1),(2,5),(3,5),(4,5)],6)=>2
([(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,5),(3,4),(4,5)],6)=>2
([(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(2,4),(2,5),(3,4),(3,5)],6)=>2
([(0,5),(1,5),(2,4),(3,4)],6)=>1
([(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,3),(3,4),(4,5)],6)=>2
([(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,4),(3,4),(4,5)],6)=>2
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6)=>2
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)=>2
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>3
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6)=>3
([(0,5),(1,4),(2,3)],6)=>1
([(1,5),(2,4),(3,4),(3,5)],6)=>1
([(0,1),(2,5),(3,4),(4,5)],6)=>1
([(1,2),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,4),(2,3),(3,5),(4,5)],6)=>2
([(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>2
([(0,1),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>2
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>2
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6)=>2
([(1,4),(1,5),(2,3),(2,5),(3,4)],6)=>2
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>2
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6)=>2
([(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,5),(1,4),(2,3),(2,4),(3,5)],6)=>1
([(0,1),(2,4),(2,5),(3,4),(3,5)],6)=>2
([(0,5),(1,5),(2,3),(2,4),(3,4)],6)=>2
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)=>2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6)=>2
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6)=>2
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,5),(4,5)],6)=>2
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,5),(4,5)],6)=>2
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(0,1),(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>2
([(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
([(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>2
([(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
([(0,5),(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),(4,5)],6)=>3
([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6)=>3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>3
([(0,5),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6)=>3
([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6)=>2
([(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5)],6)=>2
([(0,5),(1,2),(1,3),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
([(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)=>4
([(0,4),(0,5),(1,2),(1,3),(2,3),(4,5)],6)=>2
([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6)=>2
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(4,5)],6)=>2
([(0,1),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6)=>2
([(0,1),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
([(0,1),(0,5),(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),(2,5),(3,5),(4,5)],6)=>3
([(0,5),(1,2),(1,3),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>3
([(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
([(0,5),(1,2),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)=>4
([(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)=>4
([(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)=>5
([],7)=>0
([(5,6)],7)=>1
([(4,6),(5,6)],7)=>1
([(3,6),(4,6),(5,6)],7)=>2
([(2,6),(3,6),(4,6),(5,6)],7)=>2
([(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,6),(4,6),(5,6)],7)=>2
([(3,6),(4,5)],7)=>1
([(3,6),(4,5),(5,6)],7)=>1
([(2,3),(4,6),(5,6)],7)=>1
([(4,5),(4,6),(5,6)],7)=>2
([(2,6),(3,6),(4,5),(5,6)],7)=>2
([(1,2),(3,6),(4,6),(5,6)],7)=>2
([(3,6),(4,5),(4,6),(5,6)],7)=>2
([(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>2
([(0,1),(2,6),(3,6),(4,6),(5,6)],7)=>2
([(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,6),(4,5),(5,6)],7)=>2
([(1,6),(2,6),(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)=>2
([(3,5),(3,6),(4,5),(4,6)],7)=>2
([(1,6),(2,6),(3,5),(4,5)],7)=>1
([(2,6),(3,4),(3,5),(4,6),(5,6)],7)=>2
([(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,5),(4,5)],7)=>2
([(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>2
([(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>2
([(1,6),(2,6),(3,4),(3,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,4),(4,5),(5,6)],7)=>2
([(2,6),(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)=>2
([(0,6),(1,6),(2,6),(3,5),(4,5),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>2
([(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)=>2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6)],7)=>2
([(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)=>2
([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(0,6),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(0,6),(1,5),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(0,5),(0,6),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(1,6),(2,5),(3,4)],7)=>1
([(2,6),(3,5),(4,5),(4,6)],7)=>1
([(1,2),(3,6),(4,5),(5,6)],7)=>1
([(0,3),(1,2),(4,6),(5,6)],7)=>1
([(2,3),(4,5),(4,6),(5,6)],7)=>2
([(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>2
([(0,1),(2,6),(3,6),(4,5),(5,6)],7)=>2
([(2,5),(3,4),(3,6),(4,6),(5,6)],7)=>2
([(1,2),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,5),(3,4),(4,6),(5,6)],7)=>2
([(1,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>2
([(0,1),(2,6),(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)=>2
([(2,5),(2,6),(3,4),(3,6),(4,5)],7)=>2
([(1,6),(2,5),(3,4),(3,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6)],7)=>2
([(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)=>2
([(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>2
([(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,5),(3,4),(3,6),(4,5),(5,6)],7)=>2
([(0,6),(1,6),(2,3),(3,5),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,3),(2,5),(3,4),(4,6),(5,6)],7)=>2
([(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,5),(3,4),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(1,6),(2,5),(3,4),(3,5),(4,6)],7)=>1
([(1,2),(3,5),(3,6),(4,5),(4,6)],7)=>2
([(0,6),(1,5),(2,4),(3,4),(5,6)],7)=>1
([(1,6),(2,6),(3,4),(3,5),(4,5)],7)=>2
([(1,5),(2,3),(2,4),(3,6),(4,6),(5,6)],7)=>2
([(0,6),(1,4),(2,3),(3,6),(4,5),(5,6)],7)=>2
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>2
([(1,5),(2,3),(2,6),(3,6),(4,5),(4,6)],7)=>2
([(0,6),(1,3),(2,3),(4,5),(4,6),(5,6)],7)=>2
([(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,5),(2,5),(3,4),(4,6),(5,6)],7)=>2
([(0,1),(2,6),(3,5),(4,5),(4,6),(5,6)],7)=>2
([(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)=>2
([(0,5),(1,6),(2,3),(2,4),(3,6),(4,6),(5,6)],7)=>2
([(0,5),(1,6),(2,3),(2,6),(3,6),(4,5),(4,6)],7)=>2
([(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)=>2
([(0,6),(1,5),(2,3),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,1),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,6),(5,6)],7)=>2
([(0,4),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5)],7)=>2
([(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)=>2
([(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>2
([(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,6),(1,6),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>2
([(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,6),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,6),(1,5),(2,3),(2,5),(3,6),(4,5),(4,6)],7)=>2
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(0,6),(1,5),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,5),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5)],7)=>2
([(0,1),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7)=>2
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,6)],7)=>3
([(0,6),(1,4),(2,5),(2,6),(3,5),(3,6),(4,5),(5,6)],7)=>2
([(0,6),(1,6),(2,3),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>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),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,4),(1,5),(1,6),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)=>3
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(4,6),(5,6)],7)=>3
([(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)=>3
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)=>3
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)=>3
([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7)=>2
([(0,6),(1,2),(1,3),(2,5),(3,4),(4,6),(5,6)],7)=>2
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7)=>2
([(1,6),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)=>2
([(0,4),(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7)=>2
([(0,6),(1,2),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,4),(2,3),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5)],7)=>2
([(0,5),(1,2),(1,3),(2,6),(3,6),(4,5),(4,6)],7)=>2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(5,6)],7)=>2
([(0,6),(1,4),(2,5),(2,6),(3,4),(3,5),(5,6)],7)=>2
([(0,6),(1,6),(2,3),(2,4),(3,5),(4,5),(5,6)],7)=>2
([(0,4),(1,4),(2,5),(3,5),(3,6),(4,6),(5,6)],7)=>2
([(0,4),(0,5),(1,2),(1,3),(2,6),(3,6),(4,6),(5,6)],7)=>2
([(0,4),(1,4),(1,6),(2,5),(2,6),(3,5),(3,6),(5,6)],7)=>2
([(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)=>2
([(0,6),(1,5),(2,3),(2,4),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(0,6),(1,5),(2,3),(2,4),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>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),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,6),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,6),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)=>4
([(0,6),(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,6),(4,6),(5,6)],7)=>4
([(0,6),(1,5),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(0,4),(0,5),(0,6),(1,4),(1,5),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6)],7)=>4
([(0,1),(2,5),(3,4),(4,6),(5,6)],7)=>1
([(0,3),(1,2),(4,5),(4,6),(5,6)],7)=>2
([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7)=>2
([(0,1),(2,3),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,5),(1,4),(2,3),(2,6),(3,6),(4,6),(5,6)],7)=>2
([(0,5),(0,6),(1,4),(1,6),(2,3),(2,6),(3,6),(4,6),(5,6)],7)=>2
([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)=>1
([(0,1),(2,5),(2,6),(3,4),(3,6),(4,5)],7)=>2
([(0,6),(1,5),(2,3),(2,4),(3,4),(5,6)],7)=>2
([(0,6),(1,4),(2,3),(2,6),(3,5),(4,5),(5,6)],7)=>2
([(0,4),(1,3),(2,5),(2,6),(3,5),(4,6),(5,6)],7)=>2
([(0,5),(1,2),(1,4),(2,3),(3,6),(4,6),(5,6)],7)=>2
([(0,6),(1,4),(2,3),(2,5),(3,5),(4,6),(5,6)],7)=>2
([(0,5),(1,4),(1,5),(2,3),(2,6),(3,6),(4,6)],7)=>2
([(0,5),(1,4),(2,3),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,4),(1,3),(2,5),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(1,5),(1,6),(2,3),(2,4),(3,4),(5,6)],7)=>2
([(1,3),(2,5),(2,6),(3,4),(4,5),(4,6),(5,6)],7)=>2
([(0,1),(2,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)=>2
([(0,6),(1,2),(1,3),(2,3),(4,5),(4,6),(5,6)],7)=>2
([(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,6),(1,2),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,2),(1,6),(2,6),(3,4),(3,5),(4,5),(5,6)],7)=>2
([(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,1),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,6),(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,1),(0,6),(1,6),(2,3),(2,5),(3,5),(4,5),(4,6)],7)=>2
([(0,5),(1,2),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,5),(0,6),(1,2),(1,4),(2,3),(3,5),(4,6)],7)=>2
([(0,6),(1,2),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6)],7)=>2
([(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>3
([(0,6),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>3
([(0,6),(1,5),(2,3),(2,4),(2,5),(3,4),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,5),(1,2),(1,4),(1,6),(2,3),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,3),(1,4),(1,5),(2,4),(2,5),(3,6),(4,6),(5,6)],7)=>3
([(0,3),(0,4),(1,2),(1,5),(2,5),(3,6),(4,6),(5,6)],7)=>2
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)=>3
([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6),(5,6)],7)=>3
([(0,1),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,1),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,1),(1,6),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,5),(0,6),(1,2),(1,3),(2,3),(4,5),(4,6)],7)=>2
([(0,3),(1,3),(1,4),(2,5),(2,6),(4,5),(4,6),(5,6)],7)=>2
([(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,6),(1,2),(1,4),(2,4),(3,5),(3,6),(4,5),(5,6)],7)=>2
([(0,1),(0,2),(1,2),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,6),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5)],7)=>2
([(0,2),(1,2),(1,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,1),(0,4),(1,4),(2,5),(2,6),(3,5),(3,6),(4,6),(5,6)],7)=>2
([(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>3
([(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4
([(0,6),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4
([(0,6),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4
([(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)=>4
([(0,6),(1,5),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6)],7)=>3
([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>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)=>4
([(0,6),(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)=>4
([(0,4),(0,5),(1,2),(1,6),(2,6),(3,4),(3,5),(3,6),(4,5)],7)=>2
([(0,1),(0,2),(1,2),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,4),(0,5),(1,2),(1,3),(2,3),(2,6),(3,6),(4,5),(4,6),(5,6)],7)=>2
([(0,1),(0,5),(1,5),(2,3),(2,4),(2,6),(3,4),(3,6),(4,6),(5,6)],7)=>3
([(0,1),(1,6),(2,3),(2,4),(2,5),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>3
([(0,1),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4
([(0,1),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7)=>4
([(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)=>4
([(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)=>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)=>5
([(0,2),(0,3),(0,4),(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)=>5
([(0,1),(0,2),(0,3),(0,4),(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)=>6
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(1,5),(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)=>7
([(0,7),(1,2),(1,3),(1,4),(1,5),(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)=>6
([(0,7),(1,6),(2,5),(3,4)],8)=>1
([(0,3),(1,2),(4,6),(4,7),(5,6),(5,7)],8)=>2
([(0,1),(2,5),(2,6),(2,7),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)=>4
([(0,6),(0,7),(1,3),(1,4),(2,3),(2,4),(5,6),(5,7)],8)=>2
([(0,3),(1,2),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>3
([(0,1),(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)=>5
([(0,2),(0,3),(1,2),(1,3),(4,5),(4,6),(4,7),(5,6),(5,7),(6,7)],8)=>3
([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4),(5,6),(5,7),(6,7)],8)=>3
([(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
([(0,7),(1,6),(2,5),(3,5),(3,6),(3,7),(4,5),(4,6),(4,7)],8)=>3
([(0,8),(1,6),(2,6),(3,7),(4,5),(5,8),(6,7),(7,8)],9)=>2
([(0,6),(1,6),(2,7),(3,7),(4,8),(5,7),(5,8),(6,8)],9)=>2
([(0,7),(1,7),(2,6),(3,6),(4,5),(5,8),(6,8),(7,8)],9)=>2
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(2,3),(2,4),(2,5),(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)=>8
([(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
([(0,1),(0,2),(0,3),(0,4),(0,5),(0,6),(0,7),(0,8),(0,9),(1,2),(1,3),(1,4),(1,5),(1,6),(1,7),(1,8),(1,9),(2,3),(2,4),(2,5),(2,6),(2,7),(2,8),(2,9),(3,4),(3,5),(3,6),(3,7),(3,8),(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)=>9
([(0,10),(1,7),(2,7),(3,8),(4,9),(5,6),(6,10),(7,9),(8,9),(8,10)],11)=>2
([(0,10),(1,8),(2,8),(3,7),(4,7),(5,6),(6,10),(7,9),(8,9),(9,10)],11)=>2
([(0,8),(1,8),(2,9),(3,7),(4,7),(5,6),(6,10),(7,10),(8,9),(9,10)],11)=>2
([(0,10),(1,9),(2,9),(3,7),(4,8),(5,8),(6,9),(6,10),(7,8),(7,10)],11)=>2
([(0,9),(1,9),(2,8),(3,8),(4,7),(5,7),(6,9),(6,10),(7,10),(8,10)],11)=>2
([(0,10),(1,9),(2,7),(3,7),(4,8),(5,8),(6,9),(6,10),(7,9),(8,10)],11)=>2
([(0,11),(1,10),(2,9),(3,8),(4,8),(4,9),(4,10),(5,8),(5,9),(5,11),(6,8),(6,10),(6,11),(7,9),(7,10),(7,11)],12)=>2
([(0,12),(1,8),(2,8),(3,9),(4,10),(5,11),(6,7),(7,12),(8,10),(9,11),(9,12),(10,11)],13)=>2
([(0,12),(1,9),(2,9),(3,8),(4,8),(5,10),(6,7),(7,12),(8,11),(9,11),(10,11),(10,12)],13)=>2
([(0,12),(1,8),(2,8),(3,9),(4,9),(5,10),(6,7),(7,12),(8,11),(9,10),(10,11),(11,12)],13)=>2
([(0,9),(1,9),(2,10),(3,11),(4,8),(5,8),(6,7),(7,12),(8,12),(9,11),(10,11),(10,12)],13)=>2
([(0,10),(1,10),(2,9),(3,9),(4,8),(5,8),(6,7),(7,12),(8,12),(9,11),(10,11),(11,12)],13)=>2
([(0,8),(1,8),(2,9),(3,9),(4,11),(5,10),(6,7),(7,12),(8,10),(9,11),(10,12),(11,12)],13)=>2
([(0,12),(1,11),(2,11),(3,8),(4,8),(5,9),(6,10),(7,11),(7,12),(8,10),(9,10),(9,12)],13)=>2
([(0,11),(1,12),(2,12),(3,9),(4,9),(5,8),(6,8),(7,11),(7,12),(8,10),(9,10),(10,11)],13)=>2
([(0,11),(1,11),(2,9),(3,9),(4,10),(5,8),(6,8),(7,11),(7,12),(8,12),(9,10),(10,12)],13)=>2
([(0,12),(1,11),(2,9),(3,9),(4,10),(5,8),(6,8),(7,11),(7,12),(8,11),(9,10),(10,12)],13)=>2
([(0,11),(1,9),(2,9),(3,8),(4,8),(5,10),(6,10),(7,11),(7,12),(8,12),(9,12),(10,11)],13)=>2
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Description
The dimension of a graph.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
The dimension of a graph is the least integer $n$ such that there exists a representation of the graph in the Euclidean space of dimension $n$ with all vertices distinct and all edges having unit length. Edges are allowed to intersect, however.
References
[1] Erdős, P., Harary, F., Tutte, W. T. On the dimension of a graph MathSciNet:0188096
[2] https://en.wikipedia.org/wiki/Dimension_(graph_theory)
[2] https://en.wikipedia.org/wiki/Dimension_(graph_theory)
Code
"""
On the dimension of a graph, Paul Erdös, Frank Harary and William
T. Tutte
"""
dimensions = dict()
N = 7
for n in range(1,N+1):
# C_n
if n > 3:
G = graphs.CycleGraph(n)
dimensions[G.canonical_label().copy(immutable=True)] = 2
# P_n
if n > 1:
G = graphs.PathGraph(n)
dimensions[G.canonical_label().copy(immutable=True)] = 1
# K_n
G = graphs.CompleteGraph(n)
dimensions[G.canonical_label().copy(immutable=True)] = n-1
# K_n-e
if n > 2:
G.delete_edge(G.edges()[0])
dimensions[G.canonical_label().copy(immutable=True)] = n-2
# K_{2,n}
G = graphs.CompleteBipartiteGraph(2, n)
if n >= 3:
dimensions[G.canonical_label().copy(immutable=True)] = 3
elif n == 2:
dimensions[G.canonical_label().copy(immutable=True)] = 2
# K_{3,n}
G = graphs.CompleteBipartiteGraph(3, n)
if n >= 3:
dimensions[G.canonical_label().copy(immutable=True)] = 4
# Wheel
G = graphs.WheelGraph(n) # n vertices
if n in [5, 6]:
dimensions[G.canonical_label().copy(immutable=True)] = 3
elif n == 7:
dimensions[G.canonical_label().copy(immutable=True)] = 2
elif n > 7:
dimensions[G.canonical_label().copy(immutable=True)] = 3
G = graphs.CubeGraph(n) # 2^n vertices
if n > 1:
dimensions[G.canonical_label().copy(immutable=True)] = 2
def statistic(G):
global dimensions
# bridges do not change the dimension, unless we have a forest
G = G.copy(immutable=False)
bridges = list(G.bridges())
if G.num_edges() == len(bridges):
if G.num_edges() == 0:
return 0
if max(G.degree()) <= 2:
return 1
return 2
G.delete_edges(bridges)
l = G.connected_components_subgraphs()
if len(l) > 1:
statistics = [statistic(H) for H in l]
if not any(v is None for v in statistics):
return max(statistics)
l = G.blocks_and_cut_vertices()[0]
if len(l) > 1:
statistics = [statistic(G.subgraph(B)) for B in l]
if not any(v is None for v in statistics):
return max(statistics)
G = G.canonical_label().copy(immutable=True)
if G in dimensions:
return dimensions[G]
n = G.num_verts()
if G.is_clique():
return n-1
Created
Nov 20, 2020 at 18:53 by Martin Rubey
Updated
Nov 23, 2020 at 21:20 by Martin Rubey
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