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Discussiones Mathematicae Graph Theory 25(3) (2005) 355-361
DOI: https://doi.org/10.7151/dmgt.1288

ON THE p-DOMINATION NUMBER OF CACTUS GRAPHS

Mostafa Blidia and Mustapha Chellali Department of Mathematics, University of Blida B.P. 270, Blida, Algeria e-mail: mblidia@hotmail.com e-mail: mchellali@hotmail.com

Lutz Volkmann Lehrstuhl II für Mathematik, RWTH Aachen Templergraben 55, D-52056 Aachen, Germany e-mail: volkm@math2.rwth-aachen.de

Abstract

Let p be a positive integer and G = (V,E) a graph. A subset S of V is a p-dominating set if every vertex of V−S is dominated at least p times. The minimum cardinality of a p-dominating set a of G is the p-domination number γ_p(G). It is proved for a cactus graph G that γ_p(G) £ (| V|+|L_p(G)| +c(G))/2, for every positive integer p³2, where L_p(G) is the set of vertices of G of degree at most p−1 and c(G) is the number of odd cycles in G.

Keywords: p-domination number, cactus graphs.

2000 Mathematics Subject Classification: 05C69.

References

[1]	M. Blidia, M. Chellali and L. Volkmann, Some bounds on the p-domination number in trees, submitted for publication.
[2]	J.F. Fink and M.S. Jacobson, n-domination in graphs, in: Y. Alavi and A.J. Schwenk, eds, Graph Theory with Applications to Algorithms and Computer Science (Wiley, New York, 1985) 283-300.
[3]	J.F. Fink and M.S. Jacobson, On n-domination, n-dependence and forbidden subgraphs, in: Graph Theory with Applications to Algorithms and Computer Science (Wiley, New York, 1985) 301-312.
[4]	T.W. Haynes, S.T. Hedetniemi and P.J. Slater, Fundamentals of Domination in Graphs (Marcel Dekker, New York, 1998).
[5]	T.W. Haynes, S.T. Hedetniemi and P.J. Slater (eds), Domination in Graphs: Advanced Topics (Marcel Dekker, New York, 1998).

Received 24 March 2004
Revised 26 August 2004