WALCOM 2019: WALCOM: Algorithms and Computation pp 175-187

# Computing the Metric Dimension by Decomposing Graphs into Extended Biconnected Components

(Extended Abstract)
• Duygu Vietz
• Stefan Hoffmann
• Egon Wanke
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 11355)

## Abstract

A vertex set $$U \subseteq V$$ of an undirected graph $$G=(V,E)$$ is a resolving set for G, if for every two distinct vertices $$u,v \in V$$ there is a vertex $$w \in U$$ such that the distance between u and w and the distance between v and w are different. The 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. A lot of 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 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. Furthermore, we show that the decision problem Metric Dimension remains NP-complete when the above limitation is extended to usual biconnected components.

## Keywords

Graph algorithm Complexity Metric dimension Resolving set Biconnected component

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## Authors and Affiliations

• Duygu Vietz
• 1
• Stefan Hoffmann
• 1
• Egon Wanke
• 1
1. 1.Heinrich-Heine-University DuesseldorfDuesseldorfGermany