Computing the metric dimension by decomposing graphs into extended biconnected components

Abstract

A vertex set $U\subseteq V$ of an undirected graph $G=(V,E)$ is a $\text{resolving set}$ 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 $\text{Metric Dimension}$ 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 $\text{extended biconnected components}$ 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.