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An Improved Upper Bound on the Total Restrained Domination Number in Cubic Graphs

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Abstract

In this paper, we continue the study of total restrained domination in graphs. A set S of vertices in a graph G = (V, E) is a total restrained dominating set of G if every vertex of G is adjacent to some vertex in S and every vertex of \({V {\setminus} S}\) is adjacent to a vertex in \({V {\setminus} S}\) . The minimum cardinality of a total restrained dominating set of G is the total restrained domination number γ tr(G) of G. Jiang et al. (Graphs Combin 25:341–350, 2009) showed that if G is a connected cubic graph of order n, then γ tr(G) ≤ 13n/19. In this paper we improve this upper bound to γ tr(G) ≤ (n + 4)/2. We provide two infinite families of connected cubic graphs G with γ tr(G) = n/2, showing that our new improved bound is essentially best possible.

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Authors and Affiliations

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

    Justin Southey & Michael A. Henning

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  1. Justin Southey
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  2. Michael A. Henning
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Correspondence to Justin Southey.

Additional information

J. Southey research was supported in part by the South African National Research Foundation. M.A. Henning research was supported in part by the South African National Research Foundation and the University of Johannesburg.

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Southey, J., Henning, M.A. An Improved Upper Bound on the Total Restrained Domination Number in Cubic Graphs. Graphs and Combinatorics 28, 547–554 (2012). https://doi.org/10.1007/s00373-011-1059-5

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  • DOI: https://doi.org/10.1007/s00373-011-1059-5

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