Computing the metric dimension by decomposing graphs into extended biconnected components

TL;DR

This paper introduces an efficient algorithm for computing the metric dimension of certain graphs by decomposing them into extended biconnected components, and discusses the NP-completeness of the problem under different decompositions.

Contribution

It presents a novel decomposition approach into extended biconnected components and an efficient algorithm for graphs with bounded resolving set size per component.

Findings

Efficient algorithm for graphs with bounded extended biconnected components.

NP-completeness persists for usual biconnected components.

Decomposition approach aids in understanding metric dimension complexity.

Abstract

A vertex set $U \subseteq V$ of an undirected graph $G=(V,E)$ is a $\textit{resolving set}$ for $G$, if for every two distinct vertices $u,v \in V$ there is a vertex $w \in U$ such that the distances between $u$ and $w$ and the distance between $v$ and $w$ are different. The $\textit{Metric Dimension}$ of $G$ is the size of a smallest resolving set for $G$. Deciding whether a given graph $G$ has Metric Dimension at most $k$ for some integer $k$ is well-known to be NP-complete. Many research has been done to understand the complexity of this problem on restricted graph classes. In this paper, we decompose a graph into its so called $\textit{extended biconnected components}$ and present an efficient algorithm for computing the metric dimension for a class of graphs having a minimum resolving set with a bounded number of vertices in every extended biconnected component. Further we show that the decision problem METRIC DIMENSION remains NP-complete when the above limitation is extended to usual biconnected components.