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Identifier

St001746: Graphs ⟶ ℤ

Values

([],1) => 1

([],2) => 2

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

([],3) => 2

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

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

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

([],4) => 2

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

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

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

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

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

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

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

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

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

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

([],5) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 5

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

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

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

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

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

([(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) => 5

([(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) => 5

([(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) => 5

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

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

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

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

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

([],6) => 2

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

([(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) => 4

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

([(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) => 5

([(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) => 6

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

([(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) => 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) => 6

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

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

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

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

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

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

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

([(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) => 5

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

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

([(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) => 5

([(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) => 5

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Description

The coalition number of a graph.
This is the maximal cardinality of a set partition such that each block is either a dominating set of cardinality one, or is not a dominating set but can be joined with a second block to form a dominating set.

References

[1] Bakhshesh, D., Pradhan, D. On the coalition number of graphs arXiv:2111.08945

Code

def is_coalition_partition(G, s):
    for i in range(len(s)):
        p = list(s[i])
        if G.is_dominating(p) and len(p) > 1:
            return False
        if (not G.is_dominating(p) and
            not any(G.is_dominating(p + list(s[j]))
                    for j in range(len(s)))):
            return False
    return True

def statistic(G):
    n = G.num_verts()
    for k in range(n, 0, -1):
        for s in SetPartitions(G.vertices(), k):
            if is_coalition_partition(G, s):
                return k

Created

Nov 18, 2021 at 15:30 by Martin Rubey

Updated

Nov 18, 2021 at 15:30 by Martin Rubey