  A vertex set $U \subseteq V$ of an undirected graph $G=(V,E)$ is a
\textit{resolving set} for $G$ if for every two distinct vertices $u,v \in V$
there is a vertex $w \in U$ such that the distance between $u$ and $w$ and the
distance between $v$ and $w$ are different. A resolving set $U$ is {\em
fault-tolerant} if for every vertex $u\in U$ set $U\setminus \{u\}$ is still a
resolving set. {The \em (fault-tolerant) Metric Dimension} of $G$ is the size
of a smallest (fault-tolerant) resolving set for $G$. The {\em weighted
(fault-tolerant) Metric Dimension} for a given cost function $c: V
\longrightarrow \mathbb{R}_+$ 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.
