DMGT

ISSN 1234-3099 (print version)

ISSN 2083-5892 (electronic version)

https://doi.org/10.7151/dmgt

Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory

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Discussiones Mathematicae Graph Theory 25(3) (2005) 355-361
DOI: 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 γp(G). It is proved for a cactus graph G that γp(G) £ (| V|+|Lp(G)| +c(G))/2, for every positive integer p³2, where Lp(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