Identifier
Values
([(0,1)],2) => ([],1) => ([],1) => 0
([(1,2)],3) => ([],1) => ([],1) => 0
([(0,2),(1,2)],3) => ([(0,1)],2) => ([],2) => 0
([(0,1),(0,2),(1,2)],3) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(2,3)],4) => ([],1) => ([],1) => 0
([(1,3),(2,3)],4) => ([(0,1)],2) => ([],2) => 0
([(0,3),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,3),(1,2)],4) => ([],2) => ([(0,1)],2) => 0
([(0,3),(1,2),(2,3)],4) => ([(0,2),(1,2)],3) => ([(1,2)],3) => 0
([(1,2),(1,3),(2,3)],4) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(2,3)],4) => 0
([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,3),(1,2)],4) => 0
([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3)],5) => 0
([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,5),(1,4),(2,3)],6) => 0
([(3,4)],5) => ([],1) => ([],1) => 0
([(2,4),(3,4)],5) => ([(0,1)],2) => ([],2) => 0
([(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,4),(1,4),(2,4),(3,4)],5) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 0
([(1,4),(2,3)],5) => ([],2) => ([(0,1)],2) => 0
([(1,4),(2,3),(3,4)],5) => ([(0,2),(1,2)],3) => ([(1,2)],3) => 0
([(0,1),(2,4),(3,4)],5) => ([(1,2)],3) => ([(0,2),(1,2)],3) => 0
([(2,3),(2,4),(3,4)],5) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(0,4),(1,4),(2,3),(3,4)],5) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => 0
([(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(2,3)],4) => 0
([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(2,4),(3,4)],5) => 0
([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,3),(1,2)],4) => 0
([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 0
([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3)],5) => 0
([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 0
([(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 0
([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => 0
([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 0
([(0,1),(2,3),(2,4),(3,4)],5) => ([(1,2),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 0
([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 0
([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 0
([(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) => 1
([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 1
([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7) => 1
([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(1,2),(1,3),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 0
([(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,2),(0,3),(0,4),(0,5),(1,2),(1,3),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,5),(1,4),(2,3)],6) => 0
([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,5),(1,6),(2,3),(2,4),(2,6),(3,4),(3,5),(4,5),(4,6),(5,6)],7) => ([(0,5),(1,4),(2,3),(3,6),(4,6),(5,6)],7) => 0
([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => ([(0,1),(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,6),(4,5),(5,6)],7) => ([(0,3),(0,6),(1,2),(1,6),(2,5),(3,5),(4,5),(4,6)],7) => 1
([(4,5)],6) => ([],1) => ([],1) => 0
([(3,5),(4,5)],6) => ([(0,1)],2) => ([],2) => 0
([(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([],4) => 0
([(0,5),(1,5),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([],5) => 0
([(2,5),(3,4)],6) => ([],2) => ([(0,1)],2) => 0
([(2,5),(3,4),(4,5)],6) => ([(0,2),(1,2)],3) => ([(1,2)],3) => 0
([(1,2),(3,5),(4,5)],6) => ([(1,2)],3) => ([(0,2),(1,2)],3) => 0
([(3,4),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,2)],3) => ([],3) => 0
([(1,5),(2,5),(3,4),(4,5)],6) => ([(0,3),(1,2),(1,3),(2,3)],4) => ([(1,3),(2,3)],4) => 0
([(0,1),(2,5),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 0
([(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(1,2),(1,3),(2,3)],4) => ([(2,3)],4) => 0
([(0,5),(1,5),(2,5),(3,4),(4,5)],6) => ([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,4),(3,4)],5) => 0
([(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) => ([(2,4),(3,4)],5) => 0
([(0,5),(1,5),(2,5),(3,4),(3,5),(4,5)],6) => ([(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) => ([(2,5),(3,5),(4,5)],6) => 0
([(2,4),(2,5),(3,4),(3,5)],6) => ([(0,2),(0,3),(1,2),(1,3)],4) => ([(0,3),(1,2)],4) => 0
([(0,5),(1,5),(2,4),(3,4)],6) => ([(0,3),(1,2)],4) => ([(0,2),(0,3),(1,2),(1,3)],4) => 0
([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4)],5) => 0
([(0,5),(1,5),(2,3),(3,4),(4,5)],6) => ([(0,3),(1,2),(1,4),(2,4),(3,4)],5) => ([(0,4),(1,2),(1,3),(2,4),(3,4)],5) => 0
([(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => ([(1,4),(2,3)],5) => 0
([(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => ([(1,4),(2,3),(3,4)],5) => 0
([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => ([(1,3),(1,4),(2,3),(2,4)],5) => 0
([(0,5),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => 0
([(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,4),(3,4),(3,5)],6) => 0
([(0,5),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,5),(1,2),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 0
([(0,5),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,1),(0,5),(0,6),(1,4),(1,6),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,6),(2,5),(3,5),(3,6),(4,5),(4,6)],7) => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,3),(0,4),(0,5),(1,2),(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,5),(3,4)],6) => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6) => ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6) => 0
([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => ([(0,1),(0,2),(0,6),(1,2),(1,5),(2,4),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,5),(0,6),(1,4),(1,6),(2,4),(2,5),(3,4),(3,5),(3,6)],7) => 0
([(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(0,5),(0,6),(1,2),(1,4),(1,5),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,6),(5,6)],7) => ([(1,5),(1,6),(2,3),(2,4),(3,6),(4,5)],7) => 0
([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(0,6),(1,2),(1,3),(1,6),(2,3),(2,5),(2,6),(3,4),(3,6),(4,5),(4,6),(5,6)],7) => ([(1,4),(1,6),(2,3),(2,6),(3,5),(4,5),(5,6)],7) => 0
([(0,5),(1,4),(2,3)],6) => ([],3) => ([(0,1),(0,2),(1,2)],3) => 0
([(1,5),(2,4),(3,4),(3,5)],6) => ([(0,3),(1,2),(2,3)],4) => ([(0,3),(1,2),(2,3)],4) => 0
([(0,1),(2,5),(3,4),(4,5)],6) => ([(1,3),(2,3)],4) => ([(0,3),(1,2),(1,3),(2,3)],4) => 0
([(1,2),(3,4),(3,5),(4,5)],6) => ([(1,2),(1,3),(2,3)],4) => ([(0,3),(1,3),(2,3)],4) => 0
([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 0
([(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(1,2),(1,3),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(3,4)],5) => 0
([(0,1),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(1,4),(2,3),(2,4),(3,4)],5) => ([(0,4),(1,4),(2,3),(2,4),(3,4)],5) => 0
([(0,4),(1,5),(2,3),(2,5),(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),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0
([(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,5),(2,4),(3,4),(4,5)],6) => 0
([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(0,6),(1,3),(1,4),(2,3),(2,4),(2,5),(2,6),(3,4),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,6),(2,5),(3,5),(4,5),(4,6),(5,6)],7) => 0
([(1,4),(1,5),(2,3),(2,5),(3,4)],6) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,3)],5) => 1
([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 1
([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(1,2),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 1
([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => ([(0,1),(0,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(2,5),(3,4)],6) => 1
([(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) => ([(0,5),(1,4),(2,3),(3,5),(4,5)],6) => 0
([(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) => ([(0,5),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => 1
([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,5),(1,4),(1,5),(1,6),(2,3),(2,4),(2,6),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(1,5),(2,5),(2,6),(3,4),(3,5),(3,6),(4,6)],7) => 1
([(0,5),(1,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,6),(1,3),(1,5),(1,6),(2,3),(2,4),(2,5),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,4),(2,3),(2,5),(3,6),(4,5),(4,6),(5,6)],7) => 1
([(1,4),(1,5),(2,3),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,4),(0,6),(1,2),(1,4),(1,5),(2,3),(2,5),(2,6),(3,5),(3,6),(4,5),(4,6),(5,6)],7) => ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6),(5,6)],7) => 1
([(0,5),(1,4),(2,3),(2,4),(3,5)],6) => ([(0,4),(1,3),(2,3),(2,4)],5) => ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 0
([(0,1),(2,4),(2,5),(3,4),(3,5)],6) => ([(1,3),(1,4),(2,3),(2,4)],5) => ([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 0
([(0,5),(1,5),(2,3),(2,4),(3,4)],6) => ([(0,1),(2,3),(2,4),(3,4)],5) => ([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 0
([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,4),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,4),(0,5),(1,3),(1,5),(2,3),(2,4),(3,5),(4,5)],6) => 0
([(0,4),(1,2),(1,5),(2,5),(3,4),(3,5)],6) => ([(0,2),(1,4),(1,5),(2,3),(3,4),(3,5),(4,5)],6) => ([(0,3),(0,5),(1,3),(1,5),(2,4),(2,5),(3,4),(4,5)],6) => 0
([(0,4),(1,2),(2,5),(3,4),(3,5),(4,5)],6) => ([(0,4),(1,3),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 0
([(0,1),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => ([(1,3),(1,4),(1,5),(2,3),(2,4),(2,5),(3,5),(4,5)],6) => ([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 0
([(0,4),(1,4),(2,3),(2,5),(3,5),(4,5)],6) => ([(0,1),(0,5),(1,5),(2,3),(2,4),(3,4),(3,5),(4,5)],6) => ([(0,5),(1,3),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 0
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Description
The minimal number of occurrences of the comparability-pattern in a linear ordering of the vertices of the graph.
A graph is a comparability graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges and $(a,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
A graph is a comparability graph if and only if in any linear ordering of its vertices, there are no three vertices $a < b < c$ such that $(a,b)$ and $(b,c)$ are edges and $(a,c)$ is not an edge. This statistic is the minimal number of occurrences of this pattern, in the set of all linear orderings of the vertices.
Map
line graph
Description
The line graph of a graph.
Let $G$ be a graph with edge set $E$. Then its line graph is the graph with vertex set $E$, such that two vertices $e$ and $f$ are adjacent if and only if they are incident to a common vertex in $G$.
Let $G$ be a graph with edge set $E$. Then its line graph is the graph with vertex set $E$, such that two vertices $e$ and $f$ are adjacent if and only if they are incident to a common vertex in $G$.
Map
complement
Description
The complement of a graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.
The complement of a graph has the same vertices, but exactly those edges that are not in the original graph.
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