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Identifier

St001302: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 1

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

([],3) => 1

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

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

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

([],4) => 1

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

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

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

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

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

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

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

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

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

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

([],5) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 8

([(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) => 4

([(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) => 5

([(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) => 5

([(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) => 4

([(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) => 4

([(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) => 7

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

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

([],6) => 1

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 5

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

([(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) => 3

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

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

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

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

([(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) => 3

([(0,4),(0,5),(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)],6) => 10

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

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

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

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

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

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

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

([(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) => 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) => 5

([(0,5),(1,4),(2,3),(2,4),(3,5),(4,5)],6) => 7

([(1,2),(1,5),(2,4),(3,4),(3,5),(4,5)],6) => 7

([(0,5),(1,2),(1,4),(2,3),(3,5),(4,5)],6) => 7

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Description

The number of minimally dominating sets of vertices of a graph.
A subset of vertices is dominating if every vertex is either in this subset or adjacent to an element therein [1]. If a set of vertices is dominating, then so is every superset of this set. This statistic counts the minimally dominating sets.

References

[1] wikipedia:Dominating_set

Code

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

def minimal_dominating_sets(G):
    Vs = list(Subsets(G.vertices()))
    mdoms = []
    while Vs:
        V = Vs.pop(0)
        if is_dominating(G,V):
            mdoms.append(V)
            Vs = filter( lambda X: not X.issuperset(V), Vs)
    return mdoms

def statistic(G):
    return len(minimal_dominating_sets(G))

Created

Dec 10, 2018 at 15:10 by Christian Stump

Updated

Dec 10, 2018 at 15:10 by Christian Stump