Interplay between the local metric dimension and the clique number of a graph

TL;DR

This paper explores the relationship between the local metric dimension and the clique number of graphs, establishing bounds, classifications, and proving a conjecture for specific cases, while also providing a counterexample for planar graphs.

Contribution

It establishes bounds for local metric dimension based on clique number, classifies extremal graphs, and proves a conjecture for certain clique numbers, also disproving a planar graph inequality.

Findings

If clique number ≤ n-3, then local metric dimension ≤ n-3.

Graphs with local metric dimension = n-3 are classified.

Conjecture on local metric dimension bound proved for specific clique numbers.

Abstract

The local metric dimension ${\rm dim}_l$ in relation to the clique number $\omega$ is investigated. It is proved that if $\omega(G)\leq n(G)-3$, then ${\rm dim}_l(G) \leq n(G)-3$ and the graphs attaining the bound classified. Moreover, the graphs $G$ with ${\rm dim}_l(G) = n(G)-3$ are listed (with no condition on the clique number). It is proved that if $\omega(G)=n(G)-2$, then $n(G)-4 \leq {\rm dim}_l(G)\leq n(G)-3$, and all graphs are divided into two groups depending on which of the options applies. The conjecture asserting that for any graph $G$ we have ${\rm dim}_l(G) \leq \left[(\omega(G)-2)/(\omega(G)-1)\right] \cdot n(G)$ is proved for all graphs with $\omega(G)\in\{n(G)-1,n(G)-2,n(G)-3\}$. A negative answer is given for the problem whether every planar graph fulfills the inequality ${\rm dim}_l(G) \leq \lceil (n(G)+1)/2 \rceil$.