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The diameter of paired-domination vertex critical graphs

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Abstract

In this paper we continue the study of paired-domination in graphs introduced by Haynes and Slater (Networks 32 (1998), 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 γ pr(G), is the minimum cardinality of a paired-dominating set of G. The graph G is paired-domination vertex critical if for every vertex v of G that is not adjacent to a vertex of degree one, γ pr(G − v) < γ pr(G). We characterize the connected graphs with minimum degree one that are paired-domination vertex critical and we obtain sharp bounds on their maximum diameter. We provide an example which shows that the maximum diameter of a paired-domination vertex critical graph is at least 3/2 (γ pr(G) − 2). For γ pr(G) ⩽ 8, we show that this lower bound is precisely the maximum diameter of a paired-domination vertex critical graph.

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Authors and Affiliations

  1. School of Mathematical Sciences, University of KwaZulu-Natal, Pietermaritzburg, 3209, South Africa

    Michael A. Henning

  2. Department of Mathematics and Statistics, University of Victoria, Victoria, BC, Canada, V8W 3P4

    Christina M. Mynhardt

Authors
  1. Michael A. Henning
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  2. Christina M. Mynhardt
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Additional information

The first author was supported in part by the South African National Research Foundation and the University of KwaZulu-Natal, the second author was supported by the Natural Sciences and Engineering Research Council of Canada.

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Henning, M.A., Mynhardt, C.M. The diameter of paired-domination vertex critical graphs. Czech Math J 58, 887–897 (2008). https://doi.org/10.1007/s10587-008-0057-0

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  • DOI: https://doi.org/10.1007/s10587-008-0057-0

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