Total Restrained Domination in Cubic Graphs

  • Authors:

  • Hongxing Jiang;Liying Kang;Erfang Shan

  • Affiliations:

  • Shanghai University, Department of Mathematics, 200444, Shanghai, P.R. China and Wenzhou University, Department of Mathematics, 325000, Wenzhou, P.R. China;Shanghai University, Department of Mathematics, 200444, Shanghai, P.R. China;Shanghai University, Department of Mathematics, 200444, Shanghai, P.R. China

  • Venue:
  • Graphs and Combinatorics
  • Year:
  • 2009

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Abstract

A set S of vertices in a graph G = (V, E) is a total restrained dominating set (TRDS) of G if every vertex of G 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 minimum cardinality of a TRDS of G. Let G be a cubic graph of order n. In this paper we establish an upper bound on γ tr (G). If adding the restriction that G is claw-free, then we show that γ tr (G) = γ t (G) where γ t (G) is the total domination number of G, and thus some results on total domination in claw-free cubic graphs are valid for total restrained domination.