Paired-domination in generalized claw-free graphs |
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Authors: | Paul Dorbec Sylvain Gravier Michael A Henning |
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Institution: | (1) ERTé “Maths à modeler”, Laboratoire Leibniz, 46 av. F. Viallet, 38031 Grenoble Cedex, France;(2) School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg, 3209, South Africa |
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Abstract: | In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998) 199–206).
A set S of vertices in a graph G is a paired-dominating set of G if every vertex of G is adjacent to some vertex in S and if the subgraph induced by S contains a perfect matching. The paired-domination number of G, denoted by , is the minimum cardinality of a paired-dominating set of G. If G does not contain a graph F as an induced subgraph, then G is said to be F-free. Haynes and Slater (Networks 32 (1998) 199–206) showed that if G is a connected graph of order , then and this bound is sharp for graphs of arbitrarily large order. Every graph is -free for some integer a ≥ 0. We show that for every integer a ≥ 0, if G is a connected -free graph of order n ≥ 2, then with infinitely many extremal graphs. |
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Keywords: | Bounds Generalized claw-free graphs Paired-domination |
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