1-movable Restrained Total Domination in Graphs

Abstract

A nonempty subset S of V (G) is a 1-movable restrained total dominating set of G if S = V (G) or S is a restrained total dominating set of G and for every v ∈ S, S \ {v} is a restrained total dominating set of G or there exists u ∈ (V (G) \ S) ∩ NG(v) such that (S \ {v}) ∪ {u} is a restrained total dominating set of G. The smallest cardinality of a 1-movable restrained total dominating set of G is called 1-movable restrained total domination number of G, denoted by γmrt 1 (G). A 1-movable restrained total dominating set of G with cardinality equal to γmrt 1 (G) is called γmrt 1 -set of G. This paper characterized 1-movable restrained total dominating sets in the join of two graphs.