On metric dimension of permutation graphs

Abstract

The metric dimension $\dim (G)$ of a graph $G$ is the minimum number of vertices such that every vertex of $G$ is uniquely determined by its vector of distances to the set of chosen vertices. Let $G_1$ and $G_2$ be disjoint copies of a graph $G$, and let $\sigma : V(G_1) \rightarrow V(G_2)$ be a permutation. Then, a permutation graph $G_{\sigma }=(V, E)$ has the vertex set $V=V(G_1) \cup V(G_2)$ and the edge set $E=E(G_1) \cup E(G_2) \cup \{uv \mid v=\sigma (u)\}$. We show that $2 \le \dim (G_{\sigma }) \le n-1$ for any connected graph $G$ of order $n$ at least $3$. We give examples showing that neither is there a function $f$ such that $\dim (G)<f(\dim (G_{\sigma }))$ for all pairs $(G,\sigma )$, nor is there a function $g$ such that $g(\dim (G))>\dim (G_{\sigma })$ for all pairs $(G, \sigma )$. Further, we characterize permutation graphs $G_{\sigma }$ satisfying $\dim (G_{\sigma })=n-1$ when $G$ is a complete $k$-partite graph, a cycle, or a path on $n$ vertices.