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.