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Identifier
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.
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)
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