An upper bound on the total restrained domination number of a tree

  • Authors:

  • Johannes H. Hattingh;Elizabeth Jonck;Ernst J. Joubert

  • Affiliations:

  • Department of Mathematics and Statistics, University Plaza, Georgia State University, Atlanta, USA 30303;Department of Mathematics, University of Johannesburg, Auckland Park, South Africa 2006;Department of Mathematics, University of Johannesburg, Auckland Park, South Africa 2006

  • Venue:
  • Journal of Combinatorial Optimization
  • Year:
  • 2010

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Abstract

Let G=(V,E) be a graph. A set of vertices S驴V is a total restrained dominating set if every vertex is adjacent to a vertex in S and every vertex of $V-\nobreak S$ is adjacent to a vertex in V驴S. The total restrained domination number of G, denoted by 驴 tr (G), is the smallest cardinality of a total restrained dominating set of G. A support vertex of a graph is a vertex of degree at least two which is adjacent to a leaf. We show that $\gamma_{\mathit{tr}}(T)\leq\lfloor\frac{n+2s+\ell-1}{2}\rfloor$ where T is a tree of order n驴3, and s and 驴 are, respectively, the number of support vertices and leaves of T. We also constructively characterize the trees attaining the aforementioned bound.