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

Author

Listed:
  • Paul Dorbec

    (Université de Bordeaux – CNRS)

  • Bert Hartnell

    (Saint Mary’s University)

  • Michael A. Henning

    (University of Johannesburg)

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 $\gamma_{\mathrm{pr}}(G)\le ( \frac{2r^{2}-6r+6}{r(r-1)} )\gamma_{\times2}(G)$ .

Suggested Citation

  • Paul Dorbec & Bert Hartnell & Michael A. Henning, 2014. "Paired versus double domination in K 1,r -free graphs," Journal of Combinatorial Optimization, Springer, vol. 27(4), pages 688-694, May.
    • Handle: RePEc:spr:jcomop:v:27:y:2014:i:4:d:10.1007_s10878-012-9547-y
    DOI: 10.1007/s10878-012-9547-y
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    References listed on IDEAS

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    1. Paul Dorbec & Sylvain Gravier & Michael A. Henning, 2007. "Paired-domination in generalized claw-free graphs," Journal of Combinatorial Optimization, Springer, vol. 14(1), pages 1-7, July.
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