On the global offensive alliance number of a graph release_npkvr57tmrfkzlxxyxq2zv2ecq

by J. M. Sigarreta, J. A. Rodriguez

Released as a article .

2006  

Abstract

An offensive alliance in a graph Γ=(V,E) is a set of vertices S⊂ V where for every vertex v in its boundary it holds that 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 Γ. The offensive alliance number a_o(Γ) (respectively, strong offensive alliance number a_ô(Γ)) is the minimum cardinality of an offensive (respectively, strong offensive) alliance in Γ. The global offensive alliance number γ_o(Γ) and the global strong offensive alliance number γ_ô(Γ) are defined similarly. Clearly, a_o(Γ)<γ_o(Γ) and a_ô(Γ)<γ_ô(Γ). It was shown in [Discuss. Math. Graph Theory, 24 (2004), no. 2, 263-275] that a_o(Γ)<2n/3 and a_ô(Γ)<5n/6, where n denotes the order of Γ. In this paper we obtain several tight bounds on γ_o(Γ) and γ_ô(Γ) in terms of several parameters of Γ. For instance, we show that 2m+n/3Δ+1<γ_o(Γ)<2n/3 and 2(m+n)/3Δ+2<γ_ô(Γ)<5n/6, where m denotes the size of Γ and Δ its maximum degree (the last upper bound holds true for all Γ with minimum degree greatest or equal to two).
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Date   2006-02-20
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arXiv  math/0602432v1
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