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total restrained domination number; Nordhaus-Gaddum-type results; NP-complete; level decomposition
In this paper we initiate the study of total restrained domination in graphs. Let $G=(V,E)$ be a graph. A total restrained dominating set is a set $S\subseteq V$ where every vertex in $V-S$ is adjacent to a vertex in $S$ as well as to another vertex in $V-S$, and every vertex in $S$ is adjacent to another vertex in $S$. The total restrained domination number of $G$, denoted by $\gamma _r^t(G)$, is the smallest cardinality of a total restrained dominating set of $G$. First, some exact values and sharp bounds for $\gamma _r^t(G)$ are given in Section 2. Then the Nordhaus-Gaddum-type results for total restrained domination number are established in Section 3. Finally, we show that the decision problem for $\gamma _r^t(G)$ is NP-complete even for bipartite and chordal graphs in Section 4.
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