On the strong metric dimension of composed graphs

TL;DR

This paper investigates the strong metric dimension of composed graphs, establishing a relationship between the problem's complexity on the entire graph and its biconnected components, with implications for efficient computation.

Contribution

It demonstrates that computing a minimum strong resolving set for an undirected graph reduces to computing such sets for its biconnected components, linking their complexities.

Findings

Efficient computation of strong resolving sets depends on biconnected components.

The problem's complexity is characterized by the components' properties.

Provides a framework for analyzing the strong metric dimension in composed graphs.

Abstract

Two vertices $u$ and $v$ of an undirected graph $G$ are strongly resolved by a vertex $w$ if there is a shortest path between $w$ and $u$ containing $v$ or a shortest path between $w$ and $v$ containing $u$. A vertex set $R$ is a strong resolving set for $G$ if for each pair of vertices there is a vertex in $R$ that strongly resolves them. The strong metric dimension of $G$ is the size of a minimum strong resolving set for $G$. We show that a minimum strong resolving set for an undirected graph $G$ can be computed efficiently if and only if a minimum strong resolving set for each biconnected component of $G$ can be computed efficiently.