Opuscula Math. 30, no. 3 (2010), 249-254
http://dx.doi.org/10.7494/OpMath.2010.30.3.249

 
Opuscula Mathematica

Trees with equal global offensive k-alliance and k-domination numbers

Mustapha Chellali

Abstract. Let \(k \geq 1\) be an integer. A set \(S\) of vertices of a graph \(G = (V(G),E(G))\) is called a global offensive \(k\)-alliance if \(|N(v) \cap S| \geq |N(v) - S| + k\) for every \(v \in V(G)- S\), where \(N(v)\) is the neighborhood of \(v\). The subset \(S\) is a \(k\)-dominating set of \(G\) if every vertex in \(V(G) - S\) has at least \(k\) neighbors in \(S\). The global offensive \(k\)-alliance number \(\gamma_0^k (G)\) is the minimum cardinality of a global offensive \(k\)-alliance in \(G\) and the \(k\)-domination number \(\gamma _k (G)\) is the minimum cardinality of a \(k\)-dominating set of \(G\). For every integer \(k \geq 1\) every graph \(G\) satisfies \(\gamma_0^k (G) \geq \gamma_k (G)\). In this paper we provide for \(k \geq 2\) a characterization of trees \(T\) with equal \(\gamma_0^k (T)\) and \(\gamma_k (T)\).

Keywords: global offensive \(k\)-alliance number, \(k\)-domination number, trees.

Mathematics Subject Classification: 05C69.

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  • Mustapha Chellali
  • University of Blida, LAMDA-RO Laboratory, Department of Mathematics B.P. 270, Blida, Algeria
  • Received: 2009-08-03.
  • Revised: 2010-01-15.
  • Accepted: 2010-01-16.
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Cite this article as:
Mustapha Chellali, Trees with equal global offensive k-alliance and k-domination numbers, Opuscula Math. 30, no. 3 (2010), 249-254, http://dx.doi.org/10.7494/OpMath.2010.30.3.249

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