  The notion of resolving sets in a graph was introduced by Slater (1975) and
Harary and Melter (1976) as a way of uniquely identifying every vertex in a
graph. A set of vertices in a graph is a resolving set if for any pair of
vertices x and y there is a vertex in the set which has distinct distances to x
and y. A smallest resolving set in a graph is called a metric basis and its
size, the metric dimension of the graph. The problem of computing the metric
dimension of a graph is a well-known NP-hard problem and while it was known to
be polynomial time solvable on trees, it is only recently that efforts have
been made to understand its computational complexity on various restricted
graph classes. In recent work, Foucaud et al. (2015) showed that this problem
is NP-complete even on interval graphs. They complemented this result by also
showing that it is fixed-parameter tractable (FPT) parameterized by the metric
dimension of the graph. In this work, we show that this FPT result can in fact
be extended to all graphs of bounded tree-length. This includes well-known
classes like chordal graphs, AT-free graphs and permutation graphs. We also
show that this problem is FPT parameterized by the modular-width of the input
graph.
