Restrained Weakly Connected 2-Domination and the Lexicographic Product of Graphs

Abstract

Let G=VG,EG be a nontrivial connected graph. A subset SVG is a restrained weakly connected 2-dominating set (RWC2D-set) in G if every vertex vVG\S is adjacent to at least two vertices in S and is adjacent to another vertex in VG∖S and the subgraph ⟨S⟩w weakly induced by S is connected. A subgraph is said to be weakly induced by a set S if it includes all vertices in S and all edges in the original graph G that are connected to at least one vertex in S. The minimum cardinality of a restrained weakly connected 2-dominating set, denoted by r2wG, is called the restrained weakly connected 2-domination number. Leveraging the concept of weakly connected 2-domination by examining another parameter called restrained domination is the prime focus of this study. The newly defined parameter is explored, establishing improved upper bounds and providing conditions for graph G to admit an RWC2D-set. Additionally, sufficient conditions for the RWC2D-set on the lexicographic product of graphs were provided. It is shown that for any graph G of order n4, r2wGn−2. Moreover, the upper bound for the restrained weakly connected 2-domination number of the lexicographic product of graphs is provided.