On the local metric dimension of $K_4$-free graphs

TL;DR

This paper confirms a conjecture relating local metric dimension and clique number for $K_4$-free graphs, specifically when the clique number is 3, and addresses a related problem for planar graphs.

Contribution

It proves the conjecture for graphs with clique number 3, advancing understanding of local metric dimension bounds in such graphs.

Findings

Confirmed the conjecture for $oldsymbol{oldsymbol{oldsymbol{ ext{clique number}}=3}}$.

Resolved a related problem for planar graphs.

Established bounds on local metric dimension for $K_4$-free graphs.

Abstract

Let $G$ be a graph of order $ n(G) $, local metric dimension $ \dim_l(G) $, and clique number $ \omega(G) $. It has been conjectured that if $ n(G) \geq \omega(G) + 1 \geq 4 $, then $ \dim_l(G) \leq \left( \frac{\omega(G) - 2}{\omega(G) - 1} \right) n(G) $. In this paper the conjecture is confirmed for the case $ \omega(G) = 3 $. Consequently, a problem regarding the local metric dimension of planar graphs is also resolved.