On Distance and Strong Metric Dimension of the Modular Product

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