The fractional strong metric dimension in three graph products

Abstract

For any two distinct vertices $x$ and $y$ of a graph $G$, let $S\{x, y\}$ denote the set of vertices $z$ such that either $x$ lies on a $y-z$ geodesic or $y$ lies on an $x-z$ geodesic. Let $g: V(G) \rightarrow [0,1]$ be a real valued function and, for any $U \subseteq V(G)$, let $g(U)=\sum_{v \in U}g(v)$. The function $g$ is a strong resolving function of $G$ if $g(S\{x, y\}) \ge 1$ for every pair of distinct vertices $x, y$ of $G$. The fractional strong metric dimension, $sdim_f(G)$, of a graph $G$ is $\min\{g(V(G)): g \mbox{ is a strong resolving function of }G\}$. In this paper, after obtaining some new results for all connected graphs, we focus on the study of the fractional strong metric dimension of the corona product, the lexicographic product, and the Cartesian product of graphs.