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Graphs with large paired-domination number

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  • Michael A. Henning

    (University of KwaZulu-Natal)

Abstract

In this paper, we continue the study of paired-domination in graphs introduced by Haynes and Slater (1998) Networks 32: 199–206. A paired-dominating set of a graph G with no isolated vertex is a dominating set of vertices whose induced subgraph has 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. Let G be a connected graph of order n with minimum degree at least two. Haynes and Slater (1998) Networks 32: 199–206, showed that if n ≥ 6, then $$\gamma_{\rm pr}(G) \le 2n/3$$ . In this paper, we show that there are exactly ten graphs that achieve equality in this bound. For n ≥ 14, we show that $$\gamma_{\rm pr}(G) \le 2(n-1)/3$$ and we characterize the (infinite family of) graphs that achieve equality in this bound.

Suggested Citation

  • Michael A. Henning, 2007. "Graphs with large paired-domination number," Journal of Combinatorial Optimization, Springer, vol. 13(1), pages 61-78, January.
    • Handle: RePEc:spr:jcomop:v:13:y:2007:i:1:d:10.1007_s10878-006-9014-8
    DOI: 10.1007/s10878-006-9014-8
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    References listed on IDEAS

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    1. 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.
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    Cited by:

    1. Wei Yang & Xinhui An & Baoyindureng Wu, 2017. "Paired-domination number of claw-free odd-regular graphs," Journal of Combinatorial Optimization, Springer, vol. 33(4), pages 1266-1275, May.
    2. Lei Chen & Changhong Lu & Zhenbing Zeng, 2012. "Vertices in all minimum paired-dominating sets of block graphs," Journal of Combinatorial Optimization, Springer, vol. 24(3), pages 176-191, October.
    3. Teresa W. Haynes & Michael A. Henning, 2021. "Bounds on the semipaired domination number of graphs with minimum degree at least two," Journal of Combinatorial Optimization, Springer, vol. 41(2), pages 451-486, February.
    4. Henning, Michael A. & Pilśniak, Monika & Tumidajewicz, Elżbieta, 2022. "Bounds on the paired domination number of graphs with minimum degree at least three," Applied Mathematics and Computation, Elsevier, vol. 417(C).
    5. Michael A. Henning & John McCoy, 2011. "Which trees have a differentiating-paired dominating set?," Journal of Combinatorial Optimization, Springer, vol. 22(1), pages 1-18, July.
    6. Justin Southey & Michael A. Henning, 2011. "A characterization of graphs with disjoint dominating and paired-dominating sets," Journal of Combinatorial Optimization, Springer, vol. 22(2), pages 217-234, August.
    7. Lei Chen & Changhong Lu & Zhenbing Zeng, 2010. "Labelling algorithms for paired-domination problems in block and interval graphs," Journal of Combinatorial Optimization, Springer, vol. 19(4), pages 457-470, May.
    8. 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.
    9. S. L. Fitzpatrick & B. L. Hartnell, 2010. "Well paired-dominated graphs," Journal of Combinatorial Optimization, Springer, vol. 20(2), pages 194-204, August.

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    1. Lei Chen & Changhong Lu & Zhenbing Zeng, 2012. "Vertices in all minimum paired-dominating sets of block graphs," Journal of Combinatorial Optimization, Springer, vol. 24(3), pages 176-191, October.
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