On Distance and Strong Metric Dimension of the Modular Product

Abstract

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

24 pages, 1 figure, 27 references