On the Hardness of Strong Metric Dimension

Prafullkumar Tale

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arXiv · 2608.06747

On the Hardness of Strong Metric Dimension

Prafullkumar Tale

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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.

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Prafullkumar Tale. 2026-08-07. On the Hardness of Strong Metric Dimension. https://arxiv.org/abs/2608.06747

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