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Bounds on the Total Restrained Domination Number of a Graph

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Abstract

Let G = (V, E) be a graph. A set \({S \subseteq V}\) is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γ tr (G), is the smallest cardinality of a total restrained dominating set of G. We show that if δ ≥ 3, then γ tr (G) ≤ n − δ − 2 provided G is not one of several forbidden graphs. Furthermore, we show that if G is r − regular, where 4 ≤ r ≤ n − 3, then γ tr (G) ≤ n − diam(G) − r + 1.

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Authors and Affiliations

  1. Department of Mathematics and Statistics, University Plaza, Georgia State University, Atlanta, GA, 30303, USA

    J. H. Hattingh

  2. Department of Mathematics, University of Johannesburg, P.O. Box 524, Auckland Park, 2006, South Africa

    E. Jonck & E. J. Joubert

Authors
  1. J. H. Hattingh
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  2. E. Jonck
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  3. E. J. Joubert
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Correspondence to E. J. Joubert.

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Hattingh, J.H., Jonck, E. & Joubert, E.J. Bounds on the Total Restrained Domination Number of a Graph. Graphs and Combinatorics 26, 77–93 (2010). https://doi.org/10.1007/s00373-010-0894-0

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  • DOI: https://doi.org/10.1007/s00373-010-0894-0

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