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

Johannes H. Hattingh (), Elizabeth Jonck () and Ernst J. Joubert ()
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Johannes H. Hattingh: Georgia State University
Elizabeth Jonck: University of Johannesburg
Ernst J. Joubert: University of Johannesburg

Journal of Combinatorial Optimization, 2010, vol. 20, issue 3, No 1, 205-223

Abstract: Abstract Let G=(V,E) be a graph. A set of vertices S⊆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−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. 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≥3, and s and ℓ are, respectively, the number of support vertices and leaves of T. We also constructively characterize the trees attaining the aforementioned bound.

Keywords: Total; Restrained; Domination; Trees (search for similar items in EconPapers)
Date: 2010
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DOI: 10.1007/s10878-008-9204-7

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