On the upper global powerful alliance number in trees

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Document type Article

Source type Journal

ISSN 03817032

Ars CombinatoriaVolume 151, Pages 89 - 98July 2020 On the upper global powerful alliance number in trees

Dept of Mathematics, University of Boumerdes, Algeria

Abstract For a graph G = (V,E), a set D ⊆ V is a dominating set if every vertex in V — D is either in D or has a neighbor in D. A dominating set D is a global offensive alliance (resp. a global defensive alliance) if for each vertex v in V — D (resp. v in í?) at least half the vertices from the closed neighborhood of v are in C A global powerful alliance is both global defensive and global offensive. The global powerful alliance number γpo(G) is the minimum cardinality of a global powerful alliance of G. It was shown in [1] that any tree T different from a star Sij, with order n ≥ 4, l leaves and s support vertices verifies γpo(T) ≤ 4n + 1 + s/6 . In this paper, we provide a constructive characterization of all extremal trees attaining this bound