On the global offensive alliance number of a graph

Authors:
J. M. Sigarreta;J. A. Rodríguez
Affiliations:
Department of Mathematics, University Carlos III of Madrid, Avda. de la Universidad 30, 28911 Leganés (Madrid), Spain;Department of Computer Engineering and Mathematics, Rovira i Virgili University of Tarragona, Av. Països Catalans 26, 43007 Tarragona, Spain
Venue:
Discrete Applied Mathematics
Year:
2009

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Abstract

An offensive alliance in a graph @C=(V,E) is a set of vertices S@?V where for each vertex v in its boundary the majority of vertices in v's closed neighborhood are in S. In the case of strong offensive alliance, strict majority is required. An alliance S is called global if it affects every vertex in V@?S, that is, S is a dominating set of @C. The global offensive alliance number@c"o(@C) is the minimum cardinality of a global offensive alliance in @C. An offensive alliance is connected if its induced subgraph is connected. The global-connected offensive alliance number, @c"c"o(@C), is the minimum cardinality of a global-connected offensive alliance in @C. In this paper we obtain several tight bounds on @c"o(@C) and @c"c"o(@C) in terms of several parameters of @C. The case of strong alliances is studied by analogy.