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 $(g^{\mathrm{\prime }},h^{\mathrm{\prime }})$ of $G\diamond H$ are adjacent if $g=g^{\mathrm{\prime }}$ and $h{h}^{\mathrm{\prime }}\in E(H)$, or $g{g}^{\mathrm{\prime }}\in E(G)$ and $h=h^{\mathrm{\prime }}$, or $g{g}^{\mathrm{\prime }}\in E(G)$ and $h{h}^{\mathrm{\prime }}\in E(H)$, or (for $g\mathrm{\ne }{g}^{\mathrm{\prime }}$ and $h\mathrm{\ne }{h}^{\mathrm{\prime }}$) $g{g}^{\mathrm{\prime }}\mathrm{\notin }E(G)$ and $h{h}^{\mathrm{\prime }}\mathrm{\notin }E(H)$. 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.