On the Hardness of Strong Metric Dimension

Abstract

Let \(G\) be a connected simple undirected graph. A vertex \(w\) is said to \emph{strongly resolve} a pair of distinct vertices \(u, v \in V(G)\) if either there exists an isometric path (i.e.~a shortest path) from \(w\) to \(u\) that contains \(v\), or there exists an isometric path from \(w\) to \(v\) that contains \(u\). A subset \(S \subseteq V(G)\) is said to \emph{strongly resolve} \(G\) if every pair of distinct vertices of \(G\) is strongly resolved by at least one vertex in \(S\). In the \textsc{Strong Metric Dimension} problem, the input consists of a graph \(G\) and a positive integer \(k\), and the objective is to determine whether there exists a subset \(S \subseteq V(G)\) of size at most \(k\) that strongly resolves \(G\). In this article, we show that \textsc{Strong Metric Dimension} is \NP-complete even on \((i)\) graphs of diameter two, and \((ii)\) graphs of constant pathwidth and constant feedback vertex set number.