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Semipaired Domination in Claw-Free Cubic Graphs

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Abstract

A subset S of vertices in a graph G is a dominating set if every vertex in \(V(G) {\setminus } S\) is adjacent to a vertex in S. If the graph G has no isolated vertex, then a semipaired dominating set of G is a dominating set of G with the additional property that the set S can be partitioned into two element subsets such that the vertices in each subset are at most distance two apart. The semipaired domination number \(\gamma _\mathrm{pr2}(G)\) is the minimum cardinality of a semipaired dominating set of G. We show that if G is a claw-free, connected, cubic graph of order \(n \ge 10\), then \(\gamma _\mathrm{pr2}(G) \le \frac{2}{5}n\).

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Authors and Affiliations

  1. Department of Pure and Applied Mathematics, University of Johannesburg, Auckland Park, Johannesburg, 2006, South Africa

    Michael A. Henning

  2. Theoretical and Computational Science Center, Science Laboratory Building and Department of Mathematics, King Mongkut’s University of Technology Thonburi, Bangkok, Thailand

    Pawaton Kaemawichanurat

Authors
  1. Michael A. Henning
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  2. Pawaton Kaemawichanurat
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Correspondence to Michael A. Henning.

Additional information

M. A. Henning: Research supported in part by the University of Johannesburg and the South African National Research Foundation.

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Henning, M.A., Kaemawichanurat, P. Semipaired Domination in Claw-Free Cubic Graphs. Graphs and Combinatorics 34, 819–844 (2018). https://doi.org/10.1007/s00373-018-1916-6

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  • DOI: https://doi.org/10.1007/s00373-018-1916-6

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