Paired-domination in generalized claw-free graphs

Paul Dorbec (), Sylvain Gravier () and Michael A. Henning ()
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Paul Dorbec: ERTé “Maths à modeler”, Laboratoire Leibniz
Sylvain Gravier: ERTé “Maths à modeler”, Laboratoire Leibniz
Michael A. Henning: University of KwaZulu-Natal

Journal of Combinatorial Optimization, 2007, vol. 14, issue 1, No 1, 7 pages

Abstract: 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 $$\gamma_{\rm pr}(G)$$ , 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 $$n \ge 3$$ , then $$\gamma_{\rm pr}(G) \le n-1$$ and this bound is sharp for graphs of arbitrarily large order. Every graph is $$K_{1,a+2}$$ -free for some integer a ≥ 0. We show that for every integer a ≥ 0, if G is a connected $$K_{1,a+2}$$ -free graph of order n ≥ 2, then $$\gamma_{\rm pr}(G) \le 2(an + 1)/(2a+1)$$ with infinitely many extremal graphs.

Keywords: Bounds; Generalized claw-free graphs; Paired-domination (search for similar items in EconPapers)
Date: 2007
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DOI: 10.1007/s10878-006-9022-8

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