Paired versus double domination in K 1,r -free graphs

Paul Dorbec (), Bert Hartnell () and Michael A. Henning ()
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Paul Dorbec: Université de Bordeaux – CNRS
Bert Hartnell: Saint Mary’s University
Michael A. Henning: University of Johannesburg

Journal of Combinatorial Optimization, 2014, vol. 27, issue 4, No 5, 688-694

Abstract: Abstract A vertex in G is said to dominate itself and its neighbors. A subset S of vertices is a dominating set if S dominates every vertex of G. A paired-dominating set is a dominating set whose induced subgraph contains a perfect matching. The paired-domination number of a graph G, denoted by γ pr(G), is the minimum cardinality of a paired-dominating set in G. A subset S⊆V(G) is a double dominating set of G if S dominates every vertex of G at least twice. The minimum cardinality of a double dominating set of G is the double domination number γ ×2(G). A claw-free graph is a graph that does not contain K 1,3 as an induced subgraph. Chellali and Haynes (Util. Math. 67:161–171, 2005) showed that for every claw-free graph G, we have γ pr(G)≤γ ×2(G). In this paper we extend this result by showing that for r≥2, if G is a connected graph that does not contain K 1,r as an induced subgraph, then $\gamma_{\mathrm{pr}}(G)\le ( \frac{2r^{2}-6r+6}{r(r-1)} )\gamma_{\times2}(G)$ .

Keywords: Paired-domination; Double domination; Claw-free; K 1; r -free (search for similar items in EconPapers)
Date: 2014
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DOI: 10.1007/s10878-012-9547-y

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