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Identifier

St000636: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 2

([(0,1)],2) => 2

([],3) => 3

([(1,2)],3) => 3

([(0,2),(1,2)],3) => 2

([(0,1),(0,2),(1,2)],3) => 3

([],4) => 4

([(2,3)],4) => 4

([(1,3),(2,3)],4) => 3

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

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

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

([(1,2),(1,3),(2,3)],4) => 4

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

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

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

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

([],5) => 5

([(3,4)],5) => 5

([(2,4),(3,4)],5) => 4

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

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

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

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

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

([(2,3),(2,4),(3,4)],5) => 5

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([],6) => 6

([(4,5)],6) => 6

([(3,5),(4,5)],6) => 5

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

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

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

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

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

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

([(3,4),(3,5),(4,5)],6) => 6

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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Description

The hull number of a graph.
The convex hull of a set of vertices $S$ of a graph is the smallest set $h(S)$ such that for any pair $u,v\in h(S)$ all vertices on a shortest path from $u$ to $v$ are also in $h(S)$.
The hull number is the size of the smallest set $S$ such that $h(S)$ is the set of all vertices.

Code

def statistic(G):
    from sage.graphs.convexity_properties import ConvexityProperties
    return sum(ConvexityProperties(H).hull_number() for H in G.connected_components_subgraphs())

Created

Oct 25, 2016 at 23:39 by Martin Rubey

Updated

Oct 25, 2016 at 23:39 by Martin Rubey