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

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 $n(G)\ge \omega (G)+1\ge 4$, then ${\dim }_l(G)\le \left(\frac{\omega (G)-2}{\omega (G)-1}\right)n(G)$. 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.