On the Hardness of Strong Metric Dimension

2026-08-06

AI Summary

This study investigates the computational complexity of the strong metric dimension problem: given a graph \(G\) and an integer \(k\), determine whether there exists a vertex subset of size at most \(k\) that strongly resolves all pairs of vertices in \(G\). By employing graph-theoretic modeling and polynomial-time reductions, the paper establishes for the first time that this problem remains NP-complete even when restricted to graphs of diameter two or to graphs whose pathwidth and feedback vertex set number are both constant. These results significantly sharpen the understanding of the intrinsic computational hardness of the strong metric dimension problem by demonstrating its intractability under substantially more restrictive structural conditions than previously known.

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.

Problem

• Strong Metric Dimension
• NP-completeness
• graph diameter
• pathwidth
• feedback vertex set

Innovation

• Strong Metric Dimension
• NP-completeness
• diameter two
• pathwidth
• feedback vertex set