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 $(g',h')$ 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.

Figures (1)

• Figure 1: The distance from $(g,h)$ in $G\diamond H$ when $h$ is not universal and ($g$ is not universal on the left scheme and $g$ is universal on the right scheme).

Theorems & Definitions (36)

• Theorem 1
• Proposition 2
• proof
• Theorem 3
• proof
• Theorem 4
• Theorem 6
• proof
• Theorem 7
• proof