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.
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Hattingh, J.H., Jonck, E. & Joubert, E.J. An upper bound on the total restrained domination number of a tree. J Comb Optim 20, 205–223 (2010). https://doi.org/10.1007/s10878-008-9204-7
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DOI: https://doi.org/10.1007/s10878-008-9204-7