Computing the metric dimension by decomposing graphs into extended biconnected components
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A vertex set U ⊆ V of an undirected graph G=(V,E) is a resolving set for G, if for every two distinct vertices u,v ∈ V there is a vertex w ∈ U such that the distances 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. 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 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.
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