The Fault-Tolerant Metric Dimension of Cographs

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