The Fault-Tolerant Metric Dimension of Cographs

Abstract

A vertex set (formula presented) of an undirected graph G=(V,E) is a resolving set for G if for every two distinct vertices (formula presented) there is a vertex (formula presented) 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 (formula presented) set (formula presented) 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 (formula presented) 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.