The Local Resolving Dominating Set of Comb Product Graphs

The local resolving dominating set studied in this paper is a notion that combines two concepts in graph theory, the local metric dimension and dominations in graphs. Let G and H be connected graphs of orders n and m, respectively; and x a vertex in H hereafter referred to as a linkage vertex. The comb product of G and H denoted by G.H, is a graph obtained by taking one copy of G and n copies of H and attaching the i-th copy of H at the vertex x to the i-th vertex of G. In this paper, we determine the local resolving dominating set of the comb products G . Sn with two different linkage vertices, G . Kn, G . Km,n and G . Cm graphs.