On the local metric dimension of $K_5$-free graphs
Ali Ghalavand, Xueliang Li
TL;DR
This paper proves that for $K_5$-free graphs with clique number $\omega(G)=4$, the local metric dimension satisfies $\dim_l(G)\le \left\lfloor \frac{2}{3}n(G)\right\rfloor$, extending this bound to related cases $\omega(G)=2,3$ to yield $\dim_l(G)\le \left\lfloor\frac{2}{5}n(G)\right\rfloor$ and $\dim_l(G)\le \left\lfloor\frac{1}{2}n(G)\right\rfloor$, respectively, and noting sharpness for planar graphs. The approach builds on a structural decomposition into induced subgraphs $H_1$–$H_6$ and a constructive five-stage process to form a local resolving set of size at most $\frac{2}{3}n(G)$, thereby proving the main theorem. These results support the conjectured bound $\dim_l(G)\le\left(\frac{\omega(G)-2}{\omega(G)-1}\right)n(G)$ for small $\omega(G)$ and contribute to understanding the local metric dimension in $K_5$-free graphs. The bounds are shown to be tight in planar cases, highlighting the method's optimality in this graph class.
Abstract
Let \( G \) be a graph with order \( n(G) \geq 5 \), local metric dimension \( \dim_l(G) \), and clique number \( ω(G) \). In this paper, we investigate the local metric dimension of \( K_5 \)-free graphs and prove that \( \dim_l(G) \leq \lfloor\frac{2}{3}n(G)\rfloor \) when \( ω(G) = 4 \). As a consequence of this finding, along with previous publications, we establish that if \( G \) is a \( K_5 \)-free graph, then \( \dim_l(G) \leq \lfloor\frac{2}{5}n(G)\rfloor \) when \( ω(G) = 2 \), \( \dim_l(G) \leq \lfloor\frac{1}{2}n(G)\rfloor \) when \( ω(G) = 3 \), and \( \dim_l(G) \leq \lfloor\frac{2}{3}n(G)\rfloor \) when \( ω(G) = 4 \). Notably, these bounds are sharp for planar graphs. These results for graphs with a clique number less than or equal to 4 provide a positive answer to the conjecture stating that if \( n(G) \geq ω(G) + 1 \geq 4 \), then \( \dim_l(G) \leq \left( \frac{ω(G) - 2}{ω(G) - 1} \right)n(G) \).