Restrained Inverse Domination in the Lexicographic and Cartesian Products of Two Graphs

Abstract

Let G be a connected simple graph and D be a minimum dominating set of G. A dominating set S⊆V(G)∖D is called an inverse dominating set of G with respect to D. An inverse dominating set S is called a restrained inverse dominating set of G if every vertex not in S is adjacent to a vertex in S and to a vertex in V(G)∖S. The restrained inverse domination number of G, denoted by, γ_r^((-1) ) (G), is the minimum cardinality of a restrained inverse dominating set of G. A restrained inverse dominating set of cardinality γ_r^((-1) ) (G) is called γ_r^((-1) ) (G) is called γ_r^((-1) )-set. This study is an extension of an existing research on restrained inverse domination in graphs. In this paper, we characterized the restrained inverse domination in graphs under the lexicographic and Cartesian products of two graphs.