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Identifier

St001794: Graphs ⟶ ℤ

Values

([],1) => 0

([],2) => 0

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

([],3) => 0

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

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

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

([],4) => 0

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

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

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

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

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

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

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

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

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

([(0,1),(0,2),(0,3),(1,2),(1,3),(2,3)],4) => 7

([],5) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 5

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4)],5) => 7

([(0,3),(0,4),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 9

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

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

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

([(0,3),(0,4),(1,2),(1,4),(2,4),(3,4)],5) => 9

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

([(0,1),(0,4),(1,3),(2,3),(2,4),(3,4)],5) => 7

([(0,3),(0,4),(1,2),(1,4),(2,3),(2,4),(3,4)],5) => 9

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

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

([(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 7

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 11

([(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4)],5) => 9

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,4),(3,4)],5) => 11

([(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 13

([(0,1),(0,2),(0,3),(0,4),(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)],5) => 15

([],6) => 0

([(4,5)],6) => 0

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 8

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

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

([(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 9

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 15

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5),(4,5)],6) => 17

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

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

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

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

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

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

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

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

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

([(0,5),(1,4),(1,5),(2,3),(2,5),(3,5),(4,5)],6) => 9

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

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

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

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

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Description

Half the number of sets of vertices in a graph which are dominating and non-blocking.
A set of vertices $U$ in a graph is dominating, if every vertex not in $U$ is adjacent to a vertex in $U$.
A set of vertices $U$ in a graph is non-blocking, if every vertex in $U$ is adjacent to a vertex not in $U$.
Therefore, a set of vertices is non-blocking if and only if its complement is dominating. In particular, if a set of vertices is dominating and non-blocking, so is its complement.

Code

is_dominating = lambda G, V: set(G.vertices()) == set(V).union(*[G.neighbors(v) for v in V])

is_non_blocking = lambda G, B: all(any(G.has_edge(b, v) for v in set(G.vertices()).difference(B)) for b in B)

def statistic(G):
    return sum(1 for B in Subsets(G.vertices()) if is_non_blocking(G, B) and is_dominating(G, B))//2

Created

May 16, 2022 at 10:09 by Martin Rubey

Updated

May 16, 2022 at 10:09 by Martin Rubey