Total restrained domination in graphs with minimum degree two

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 kk-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)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⧹SV⧹S is adjacent to a vertex in V⧹SV⧹S. The minimum cardinality of a total restrained dominating set of G is the total restrained domination number of G  , denoted by γtr(G)γtr(G). Let G be a connected graph of order n   with minimum degree at least 2 and with maximum degree ΔΔ where Δ⩽n-2Δ⩽n-2. We prove that if n⩾4n⩾4, then γtr(G)⩽n-Δ2-1 and this bound is sharp. If we restrict G   to a bipartite graph with Δ⩾3Δ⩾3, then we improve this bound by showing that γtr(G)⩽n-23Δ-293Δ-8-79 and that this bound is sharp.