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

[BIBTex] [PDF] [PS]

GLOBAL ALLIANCES AND INDEPENDENCE IN TREES

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

Abstract

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.

References

[1] M. Blidia, M. Chellali and O. Favaron, Independence and 2-domination in trees, Austral. J. Combin. 33 (2005) 317-327.
[2] G. Chartrand and L. Lesniak, Graphs & Digraphs: Third Edition (Chapman & Hall, London, 1996).
[3] E.J. Cockayne, O. Favaron, C. Payan and A.G. Thomason, Contributions to the theory of domination, independence, and irredundance in graphs, Discrete Math. 33 (1981) 249-258, doi: 10.1016/0012-365X(81)90268-5.
[4] T.W. Haynes, S.T. Hedetniemi, and M.A. Henning, Global defensive alliances in graphs, The Electronic J. Combin. 10 (2003) R47.
[5] T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
[6] T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs: Advanced Topics (Marcel Dekker, New York, 1998).
[7] S.M. Hedetniemi, S.T. Hedetniemi and P. Kristiansen, Alliances in graphs, J. Combin. Math. Combin. Comput. 48 (2004) 157-177.

Received 4 October 2004
Revised 26 April 2006