Total restrained domination in graphs

Abstract

AbstractIn this paper, we initiate the study of a variation of standard domination, namely total restrained domination. Let G=(V,E) be a graph. A set D⊆V is a total restrained dominating set if every vertex in V−D has at least one neighbor in D and at least one neighbor in V−D, and every vertex in D has at least one neighbor in D. The total restrained domination number of G, denoted by γtr(G), is the minimum cardinality of all total restrained dominating sets of G. We determine the best possible upper and lower bounds for γtr(G), characterize those graphs achieving these bounds and find the best possible lower bounds for γtr(G)+γtr(Ḡ) where both G and Ḡ are connected