The fractional strong metric dimension in three graph products

TL;DR

This paper introduces the fractional strong metric dimension of graphs, explores its properties, and investigates its behavior in various graph products such as corona, lexicographic, and Cartesian products.

Contribution

It defines the fractional strong metric dimension and provides new results for its calculation in multiple graph product operations.

Findings

Derived bounds for fractional strong metric dimension in various graph classes

Established formulas for the fractional strong metric dimension in graph products

Extended understanding of metric properties in complex graph constructions

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.