On Distance and Strong Metric Dimension of the Modular Product

The modular product G♢ H of graphs G and H is a graph on vertex set V(G)× V(H). Two vertices (g,h) and (g',h') of G♢ H are adjacent if g=g' and hh'∈ E(H), or gg'∈ E(G) and h=h', or gg'∈ E(G) and hh'∈ E(H), or (for g≠ g' and h≠ h') gg'∉ E(G) and hh'∉ E(H). We derive the distance formula for the modular product and then describe all edges of the strong resolving graph of G♢ H. This is then used to obtain the strong metric dimension of the modular product on several, infinite families of graphs.