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Upper paired-domination in claw-free graphs

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  • Paul Dorbec

    (Université Paris Sud 11
    University of KwaZulu-Natal)

  • Michael A. Henning

    (University of KwaZulu-Natal)

Abstract

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 maximum cardinality of a minimal paired-dominating set of G is the upper paired-domination number of G, denoted by Γpr(G). We establish bounds on Γpr(G) for connected claw-free graphs G in terms of the number n of vertices in G with given minimum degree δ. We show that Γpr(G)≤4n/5 if δ=1 and n≥3, Γpr(G)≤3n/4 if δ=2 and n≥6, and Γpr(G)≤2n/3 if δ≥3. All these bounds are sharp. Further, if n≥6 the graphs G achieving the bound Γpr(G)=4n/5 are characterized, while for n≥9 the graphs G with δ=2 achieving the bound Γpr(G)=3n/4 are characterized.

Suggested Citation

  • Paul Dorbec & Michael A. Henning, 2011. "Upper paired-domination in claw-free graphs," Journal of Combinatorial Optimization, Springer, vol. 22(2), pages 235-251, August.
    • Handle: RePEc:spr:jcomop:v:22:y:2011:i:2:d:10.1007_s10878-009-9275-0
    DOI: 10.1007/s10878-009-9275-0
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    References listed on IDEAS

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    1. Michael A. Henning, 2007. "Graphs with large paired-domination number," Journal of Combinatorial Optimization, Springer, vol. 13(1), pages 61-78, January.
    2. Michael A. Henning & Michael D. Plummer, 2005. "Vertices Contained in all or in no Minimum Paired-Dominating Set of a Tree," Journal of Combinatorial Optimization, Springer, vol. 10(3), pages 283-294, November.
    Full references (including those not matched with items on IDEAS)

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