Discussiones Mathematicae Graph Theory 27(1) (2007) 19-27
doi: 10.7151/dmgt.1340

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Mustapha Chellali

Department of Mathematics
University of Blida
B.P. 270, Blida, Algeria
e-mail: mchellali@hotmail.com

Teresa W. Haynes

Department of Mathematics
East Tennessee State University
Johnson City, TN 37614, USA
e-mail: haynes@mail.etsu.edu


A global defensive (respectively, offensive) alliance in a graph G = (V,E) is a set of vertices S ⊆ V with the properties that every vertex in V−S has at least one neighbor in S, and for each vertex v in S (respectively, in V−S) at least half the vertices from the closed neighborhood of v are in S. These alliances are called strong if a strict majority of vertices from the closed neighborhood of v must be in S. For each kind of alliance, the associated parameter is the minimum cardinality of such an alliance. We determine relationships among these four parameters and the vertex independence number for trees.

Keywords: defensive alliance, offensive alliance, global alliance, domination, trees, independence number.

2000 Mathematics Subject Classification: 05C69.


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Received 4 October 2004
Revised 26 April 2006