Computing the Metric Dimension by Decomposing Graphs into Extended Biconnected Components (Extended Abstract)

A vertex set U C V of an undirected graph G = (V, E) is a resolving set for G, if for every two distinct vertices u, v E V there is a vertex w E 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.