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Your data matches 29 different statistics following compositions of up to 3 maps.
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Matching statistic: St000097
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],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)
=> 3
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2],[3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
Description
The order of the largest clique of the graph.
A clique in a graph G is a subset U⊆V(G) such that any pair of vertices in U are adjacent. I.e. the subgraph induced by U is a complete graph.
Matching statistic: St000098
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],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)
=> 3
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2],[3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,2,2,2,2,2,2,2]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,1),(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
Description
The chromatic number of a graph.
The minimal number of colors needed to color the vertices of the graph such that no two vertices which share an edge have the same color.
Matching statistic: St000093
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 3
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2],[3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,4],[2,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[2,2,2,2,2,2,2,2]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 5
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
Description
The cardinality of a maximal independent set of vertices of a graph.
An independent set of a graph is a set of pairwise non-adjacent vertices. A maximum independent set is an independent set of maximum cardinality. This statistic is also called the independence number or stability number α(G) of G.
Matching statistic: St000786
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 3
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2],[3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2],[2],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,2,2,2],[2],[4]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,4],[2,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[2,2,2,2,2,2,2,2]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 5
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
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: St001286
(load all 2 compositions to match this statistic)
(load all 2 compositions to match this statistic)
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 3
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2],[3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2],[2],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,2,2,2],[2],[4]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,4],[2,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[2,2,2,2,2,2,2,2]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 5
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
Description
The annihilation number of a graph.
For a graph on m edges with degree sequence d1≤⋯≤dn, this is the largest number k≤n such that ∑ki=1di≤m.
Matching statistic: St001337
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 3
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2],[3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2],[2],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,2,2,2],[2],[4]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,4],[2,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[2,2,2,2,2,2,2,2]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 5
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
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 K4 by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [1].
Matching statistic: St001338
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> 3
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],6)
=> ([(0,4),(1,2),(1,3),(2,5),(3,5),(4,5)],6)
=> 3
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[2],[3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> 3
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> 4
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> 2
[[1,2],[2],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,2,2,2],[2],[4]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,4],[2,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[2,2,2,2,2,2,2,2]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ? = 5
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
[[1,1,1,1,2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ? = 4
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 K4 by joining three of the four vertices of the first with three of the four vertices of the second. For bipartite graphs the two parameters always coincide [2].
Matching statistic: St001029
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],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)
=> 3
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2],[3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2],[2],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,2,2,2],[2],[4]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,4],[2,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[2,2,2,2,2,2,2,2]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,1),(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
Description
The size of the core of a graph.
The core of the graph G is the smallest graph C such that there is a graph homomorphism from G to C and a graph homomorphism from C to G.
Matching statistic: St001494
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> 3
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],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)
=> 3
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[2],[3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 2
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 3
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> 4
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 2
[[1,2],[2],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,2,2,2],[2],[4]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,4],[2,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[2,2,2,2,2,2,2,2]]
=> ([(0,8),(2,3),(3,5),(4,2),(5,7),(6,4),(7,1),(8,6)],9)
=> ([(0,8),(1,7),(2,3),(2,4),(3,5),(4,6),(5,7),(6,8)],9)
=> ([(0,1),(0,3),(0,4),(0,5),(0,6),(0,8),(1,2),(1,4),(1,5),(1,6),(1,7),(2,3),(2,4),(2,6),(2,7),(2,8),(3,5),(3,6),(3,7),(3,8),(4,5),(4,7),(4,8),(5,7),(5,8),(6,7),(6,8),(7,8)],9)
=> ? = 5
[[1,1,1,4],[2,2,4],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,2,2,2],[2,3,3],[3,4],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,4],[2,2,2,4],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,2,2,2],[2,2,3,3],[3,3,4],[4,4],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,3],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,2,2],[2,2,2,3],[3,3,3],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,3,5],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,2,2,5],[3,3,5],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,2],[2,2,2,2],[3,4,4],[4,5],[5]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1],[2,3,3,3],[3,4,4],[4,5],[5]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,4],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,5],[3,3,3,5],[4,4,5],[5,5],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,3],[2,2,2,2,3],[3,3,3,3],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,1),(0,2),(0,3),(1,5),(1,6),(2,4),(2,6),(3,4),(3,5),(4,7),(5,7),(6,7)],8)
=> ([(0,5),(0,6),(0,7),(1,2),(1,3),(1,4),(2,6),(2,7),(3,5),(3,7),(4,5),(4,6)],8)
=> ([(0,4),(0,5),(0,6),(0,7),(1,2),(1,3),(1,6),(1,7),(2,3),(2,5),(2,7),(3,4),(3,7),(4,5),(4,6),(5,6)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,3],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,4],[3,3,3,4],[4,4,4],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,4],[4,4,5],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,2],[2,2,2,2,2],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,3],[3,3,3,3],[4,4,6],[5,6],[6]]
=> ([(0,2),(0,4),(1,6),(2,5),(3,1),(3,7),(4,3),(4,5),(5,7),(7,6)],8)
=> ([(0,3),(0,7),(1,2),(1,4),(2,5),(3,6),(4,5),(4,6),(5,7),(6,7)],8)
=> ([(0,3),(0,5),(0,6),(0,7),(1,2),(1,4),(1,5),(1,6),(2,4),(2,5),(2,7),(3,4),(3,6),(3,7),(4,5),(4,7),(5,6),(6,7)],8)
=> ? = 4
[[1,1,1,1,1,1],[2,2,2,2,2],[3,3,3,6],[4,4,6],[5,6],[6]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
[[1,1,1,1,2,2,2],[3,3],[4]]
=> ([(0,5),(2,7),(3,7),(4,1),(5,6),(6,2),(6,3),(7,4)],8)
=> ([(0,5),(1,4),(2,6),(2,7),(3,6),(3,7),(4,6),(5,7)],8)
=> ([(0,1),(0,5),(0,6),(0,7),(1,4),(1,6),(1,7),(2,3),(2,4),(2,5),(2,6),(2,7),(3,4),(3,5),(3,6),(3,7),(4,5),(4,6),(5,7),(6,7)],8)
=> ? = 4
Description
The Alon-Tarsi number of a graph.
Let G be a graph with vertices {1,…,n} and edge set E. Let PG=∏i<j,(i,j)∈Exi−xj be its graph polynomial. Then the Alon-Tarsi number is the smallest number k such that PG contains a monomial with exponents strictly less than k.
Matching statistic: St001644
Values
[[1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[[2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[[2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[[1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[[1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[[1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1],[2],[3]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[1,1,1,2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[[1,1,2,2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,2,2,2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,1,1],[2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[1,1,2],[2]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[[1,2,2],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,1],[2,2]]
=> ([],1)
=> ([],1)
=> ([],1)
=> 0 = 1 - 1
[[1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1],[5]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1,1,4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1,1],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,2],[4]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,4],[2]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],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)
=> ? = 3 - 1
[[1],[2],[4]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[[1],[3],[4]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[2],[3],[4]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1,1,1,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,1,2,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,1,1],[3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[[1,1,2],[3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,1,3],[2]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,1,3],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1,2,2],[3]]
=> ([(0,3),(2,1),(3,2)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> ([(0,3),(1,2),(2,3)],4)
=> 1 = 2 - 1
[[1,2,3],[2]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[2,2,2],[3]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,1],[2,3]]
=> ([(0,1)],2)
=> ([(0,1)],2)
=> ([],2)
=> 0 = 1 - 1
[[1,1],[3,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,2],[2,3]]
=> ([(0,2),(2,1)],3)
=> ([(0,2),(1,2)],3)
=> ([(1,2)],3)
=> 1 = 2 - 1
[[1,2],[3,3]]
=> ([(0,3),(1,4),(2,4),(3,1),(3,2)],5)
=> ([(0,4),(1,2),(1,3),(2,4),(3,4)],5)
=> ([(0,3),(1,2),(1,4),(2,4),(3,4)],5)
=> 2 = 3 - 1
[[2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],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)
=> ? = 3 - 1
[[1,6]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],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)
=> ? = 3 - 1
[[1],[6]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,5],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],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)
=> ? = 3 - 1
[[1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3],[2,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,4],[2],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,3],[3],[4]]
=> ([(0,2),(0,3),(2,5),(3,5),(4,1),(5,4)],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)
=> ? = 3 - 1
[[2,2],[3],[4]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,1,1,3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,1,3,3],[2]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2,2,2],[3]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[2,2,2,2],[3]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],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)
=> ? = 3 - 1
[[1,1,3],[3,3]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,2,2],[3,3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,3,3],[2],[3]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,1,2,2,2,2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,2,2,2,2,2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],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)
=> ? = 3 - 1
[[2,2,2,2,2,2]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 4 - 1
[[1,2,2,2,2],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,7]]
=> ([(0,6),(2,3),(3,5),(4,2),(5,1),(6,4)],7)
=> ([(0,6),(1,5),(2,3),(2,4),(3,5),(4,6)],7)
=> ([(0,1),(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)
=> ? = 4 - 1
[[1],[7]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],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)
=> ? = 3 - 1
[[1,1,6]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],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)
=> ? = 3 - 1
[[1,1],[6]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,6],[2]]
=> ([(0,5),(2,4),(3,2),(4,1),(5,3)],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)
=> ? = 3 - 1
[[1,1,1,5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,1,5],[2]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,1],[4],[5]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,2],[2],[5]]
=> ([(0,2),(0,3),(1,5),(2,4),(3,1),(3,4),(4,5)],6)
=> ([(0,3),(0,5),(1,2),(1,5),(2,4),(3,4),(4,5)],6)
=> ([(0,4),(0,5),(1,2),(1,3),(2,3),(2,5),(3,4),(4,5)],6)
=> ? = 3 - 1
[[1,5],[2],[3]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[2],[3],[4],[5]]
=> ([(0,4),(2,3),(3,1),(4,2)],5)
=> ([(0,4),(1,3),(2,3),(2,4)],5)
=> ([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5)
=> ? = 3 - 1
[[1,1,1,3],[4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
[[1,1,1,4],[3]]
=> ([(0,3),(1,5),(2,5),(3,4),(4,1),(4,2)],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)
=> ? = 3 - 1
[[1,1,1],[4,4]]
=> ([(0,4),(1,5),(2,5),(4,1),(4,2),(5,3)],6)
=> ([(0,5),(1,4),(2,4),(2,5),(3,4),(3,5)],6)
=> ([(0,1),(0,5),(1,4),(2,3),(2,4),(2,5),(3,4),(3,5),(4,5)],6)
=> ? = 4 - 1
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 following 19 statistics, ordered by result quality, also match your data. Click on any of them to see the details.
St001330The hat guessing number of a graph. St001624The breadth of a lattice. St000454The largest eigenvalue of a graph if it is integral. St001630The global dimension of the incidence algebra of the lattice over the rational numbers. St001878The projective dimension of the simple modules corresponding to the minimum of L in the incidence algebra of the lattice L. St001875The number of simple modules with projective dimension at most 1. St001877Number of indecomposable injective modules with projective dimension 2. St000771The largest multiplicity of a distance Laplacian eigenvalue in a connected graph. St000772The multiplicity of the largest distance Laplacian eigenvalue in a connected graph. St000777The number of distinct eigenvalues of the distance Laplacian of a connected graph. St001645The pebbling number of a connected graph. St000259The diameter of a connected graph. St000260The radius of a connected graph. St000302The determinant of the distance matrix of a connected graph. St000466The Gutman (or modified Schultz) index of a connected graph. St000467The hyper-Wiener index of a connected graph. St000510The number of invariant oriented cycles when acting with a permutation of given cycle type. St000681The Grundy value of Chomp on Ferrers diagrams. St000620The number of standard tableaux of shape equal to the given partition such that the minimal cyclic descent is odd.
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