The Fault-Tolerant Metric Dimension of Cographs
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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 distance between u and w and the distance between v and w are different. A resolving set U is fault-tolerant if for every vertex u∈ U set U∖{u} is still a resolving set. The (fault-tolerant) Metric Dimension of G is the size of a smallest (fault-tolerant) resolving set for G. The weighted (fault-tolerant) Metric Dimension for a given cost function c: V ⟶R_+ is the minimum weight of all (fault-tolerant) resolving sets. Deciding whether a given graph G has (fault-tolerant) Metric Dimension at most k for some integer k is known to be NP-complete. The weighted fault-tolerant Metric Dimension problem has not been studied extensively so far. In this paper we show that the weighted fault-tolerant metric dimension problem can be solved in linear time on cographs.
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