Total Restrained Domination in Cubic Graphs

A set S of vertices in a graph G = (V, E) is a total restrained dominating set (TRDS) of G if every vertex of G is adjacent to a vertex in S and every vertex of V − S is adjacent to a vertex in V − S. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of a TRDS of G. Let G be a cubic graph of order n. In this paper we establish an upper bound on γtr(G). If adding the restriction that G is claw-free, then we show that γtr(G) = γt(G) where γt(G) is the total domination number of G, and thus some results on total domination in claw-free cubic graphs are valid for total restrained domination.