  A set W \subseteq V (G) is called a resolving set, if for each pair of
distinct vertices u,v \in V (G) there exists t \in W such that d(u,t) \neq
d(v,t), where d(x,y) is the distance between vertices x and y. The cardinality
of a minimum resolving set for G is called the metric dimension of G and is
denoted by dim_M(G). A k-tree is a chordal graph all of whose maximal cliques
are the same size k + 1 and all of whose minimal clique separators are also all
the same size k. A k-path is a k-tree with maximum degree 2k, where for each
integer j, k \leq j < 2k, there exists a unique pair of vertices, u and v, such
that deg(u) = deg(v) = j. In this paper, we prove that if G is a k-path, then
dim_M(G) = k. Moreover, we provide a characterization of all 2-trees with
metric dimension two.
