In this paper, we continue the study of total restrained domination in graphs, a concept introduced by Telle and Proskurowksi (Algorithms for vertex partitioning problems on partial k-trees, SIAM J. Discrete Math. 10 (1997) 529-550) 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 \S is adjacent to a vertex in V \S. The minimum cardinality of a total restrained dominating set of G is the total restrained domination number of G, denoted by tr (G). Let G be a connected graph of order n with minimum degree at least 2 and with maximum degree where n − 2. We prove that if n 4, then tr (G) n − 2 − 1 and this bound is sharp. If we restrict G to a bipartite graph with 3, then we improve this bound by showing that tr (G) n − 2 3 − 2 9 √ 3 − 8 − 7 9 and that this bound is sharp.

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