On Distance and Strong Metric Dimension of the Modular Product

The modular product \(G\diamond H\) $G\diamond H$ of graphs G and H is a graph on vertex set \(V(G)\times V(H)\) $V(G)\times V(H)$ . Two vertices (g, h) and \((g',h')\) $(g^{\prime },h^{\prime })$ of \(G\diamond H\) $G\diamond H$ are adjacent if \(g=g'\) $g=g^{\prime }$ and \(hh'\in E(H)\) $h{h}^{\prime }\in E(H)$ , or \(gg'\in E(G)\) $g{g}^{\prime }\in E(G)$ and \(h=h'\) $h=h^{\prime }$ , or \(gg'\in E(G)\) $g{g}^{\prime }\in E(G)$ and \(hh'\in E(H)\) $h{h}^{\prime }\in E(H)$ , or (for \(g\ne g'\) $g\ne g^{\prime }$ and \(h\ne h'\) $h\ne h^{\prime }$ ) \(gg'\notin E(G)\) $g{g}^{\prime }\notin E(G)$ and \(hh'\notin E(H)\) $h{h}^{\prime }\notin E(H)$ . We derive the distance formula for the modular product and then describe all edges of the strong resolving graph of \(G\diamond H\) $G\diamond H$ . This is then used to obtain the strong metric dimension of the modular product on several, infinite families of graphs.