Restrained Disjunctive Domination in Graphs under Some Binary Operations

Abstract

A set S ⊆ V (G) is a disjunctive dominating set of a graph G if for every v ∈ V (G)\S, v is a neighbor of a vertex in S or S has at least two vertices each at distance 2 from v. We say that a disjunctive dominating set S of G is a restrained disjunctive dominating set if for each v ∈ V (G)\S there exists u ∈ V (G) \ S such that uv ∈ E(G) or there exist distinct vertices u, w ∈ V (G) \ S such that dG(u, v) = 2 = dG(w, v). The minimum cardinality γdr(G) of a restrained disjunctive dominating set of G is the restrained disjunctive domination number of G. In this paper, we characterize the restrained disjunctive dominating sets in some binary operations such as the join, corona and lexicographic product of graphs and, as a result, obtain the values of their corresponding restrained disjunctive domination numbers.