On the fractional metric dimension of corona product graphs and lexicographic product graphs

Abstract

A vertex $x$ in a graph $G$ resolves two vertices $u$, $v$ of $G$ if the distance between $u$ and $x$ is not equal to the distance between $v$ and $x$. A function $g$ from the vertex set of $G$ to $[0,1]$ is a resolving function of $G$ if $g(R_G\{u,v\})\ge 1$ for any two distinct vertices $u$ and $v$, where $R_G\{u,v\}$ is the set of vertices resolving $u$ and $v$. The real number ${\sum }_{v\in V(G)}g(v)$ is the weight of $g$. The minimum weight of all resolving functions for $G$ is called the fractional metric dimension of $G$, denoted by ${\dim }_f(G)$. In this paper we reduce the problem of computing the fractional metric dimension of corona product graphs and lexicographic product graphs, to the problem of computing some parameters of the factor graphs.