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Total restrained domination in graphs with minimum degree two. (English) Zbl 1226.05195
Summary: We continue the study of total restrained domination in graphs, a concept introduced by J. A. Telle and A. Proskurowksi [“Algorithms for vertex partitioning problems on partial $$k$$-trees,” SIAM J. Discrete Math. 10, No. 4, 529–550 (1997; Zbl 0885.68118)] as a vertex partitioning problem. A set $$S$$ of vertices in a graph $$G=(V,E)$$ is a total restrained dominating set of $$G$$ if every vertex is adjacent to a vertex in $$S$$ and every vertex of $$V\backslash S$$ is adjacent to a vertex in $$V\backslash S$$.
The minimum cardinality of a total restrained dominating set of $$G$$ is the total restrained domination number of $$G$$, denoted by $$\gamma _{\text{tr}}(G)$$. Let $$G$$ be a connected graph of order $$n$$ with minimum degree at least 2 and with maximum degree $$\varDelta$$ where $$\varDelta \leqslant n-2$$. We prove that if $$n\geqslant$$4, then $$\gamma _{\text{tr}}(G)\leqslant n-\frac{\varDelta}{2}-1$$ and this bound is sharp. If we restrict $$G$$ to a bipartite graph with $$\Delta \geqslant$$3, then we improve this bound by showing that $$\gamma _{\text{tr}}(G)\leqslant n-\frac{2}{3}\varDelta -\frac 2 9 \sqrt{3\varDelta -8} - \frac 7 9$$ and that this bound is sharp.

##### MSC:
 05C69 Vertex subsets with special properties (dominating sets, independent sets, cliques, etc.)
##### Keywords:
bounds; maximum degree; total restrained domination
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##### References:
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