Offensive alliances in cubic graphs

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Offensive alliances in graphs

A set S is an offensive alliance if for every vertex v in its boundary N(S)−S it holds that the majority of vertices in v’s closed neighbourhood are in S. The offensive alliance number is the minimum cardinality of an offensive alliance. In this paper we explore the bounds on the offensive alliance and the strong offensive alliance numbers (where a strict majority is required). In particular, w...

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Offensive r-alliances in graphs

Let G = (V , E) be a simple graph. For a nonempty set X ⊂ V , and a vertex v ∈ V , δX (v) denotes the number of neighbors v has in X . A nonempty set S ⊂ V is an offensive r-alliance in G if δS(v) ≥ δS̄(v)+ r,∀v ∈ ∂(S), where ∂(S) denotes the boundary of S. An offensive ralliance S is called global if it forms a dominating set. The global offensive r-alliance number of G, denoted by γ o r (G), i...

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Connected global offensive k-alliances in graphs

We consider finite graphs G with vertex set V (G). For a subset S ⊆ V (G), we define by G[S] the subgraph induced by S. By n(G) = |V (G)| and δ(G) we denote the order and the minimum degree of G, respectively. Let k be a positive integer. A subset S ⊆ V (G) is a connected global offensive k-alliance of the connected graphG, ifG[S] is connected and |N(v)∩S| ≥ |N(v)−S|+k for every vertex v ∈ V (G...

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On global offensive k-alliances in graphs

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Global offensive alliances in graphs and random graphs

A global o ensive alliance in a graph G = (V; E) is a subset S of V such that for every vertex v not in S at least half of the vertices in the closed neighborhood of v are in S. The cardinality of a minimum size global o ensive alliance in G is called the global o ensive alliance number of G. We give an upper bound on the global (strong) o ensive alliance number of a graph in terms of its degre...

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ژورنال

عنوان ژورنال: International Mathematical Forum

سال: 2006

ISSN: 1314-7536

DOI: 10.12988/imf.2006.06152