Strong Resolving Hop Domination in Graphs

A vertex w in a connected graph G strongly resolves two distinct vertices u and v in V(G) if v is in any shortest u -w path or if u is in any shortest v -w path. A set W of vertices in G is a strong resolving set G if every two vertices of G are strongly resolved by some vertex of W. A set S subset of V(G) is a strong resolving hop dominating set of G if S is a strong resolving set in G and for every vertex v is an element of V(G) \ S there exists u is an element of S such that dG(u, v) = 2. The smallest cardinality of such a set S is called the strong resolving hop domination number of G. This paper presents the characterization of the strong resolving hop dominating sets in the join, corona and lexicographic product of graphs. Furthermore, this paper determines the exact value or bounds of their corresponding strong resolving hop domination number.