Extremal Graph Theory for Metric Dimension and Girth

TL;DR

This paper establishes an upper bound on the metric dimension of connected graphs with cycles based on the order and girth, characterizing cases of equality for specific graph classes.

Contribution

It proves a new upper bound on the metric dimension related to girth and characterizes when equality holds for certain graph types.

Findings

Bound: 1(G)  n - g(G) + 2 for connected graphs with cycles

Equality holds iff G is a cycle, complete graph, or complete bipartite graph K_{s,t} with s,t  2

Provides a characterization of graphs achieving the bound

Abstract

A set $W\subseteq V(G)$ is called a resolving set for $G$, if for each two distinct vertices $u,v\in V(G)$ there exists $w\in W$ such that $d(u,w)\neq d(v,w)$, where $d(x,y)$ is the distance between the vertices $x$ and $y$. The minimum cardinality of a resolving set for $G$ is called the metric dimension of $G$, and denoted by $\beta(G)$. In this paper, it is proved that in a connected graph $G$ of order $n$ which has a cycle, $\beta(G)\leq n-g(G)+2$, where $g(G)$ is the length of a shortest cycle in $G$, and the equality holds if and only if $G$ is a cycle, a complete graph or a complete bipartite graph $K_{s,t}$, $ s,t\geq 2$.