  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 distances between $u$ and $w$ and
the distance between $v$ and $w$ are different. The $\textit{Metric Dimension}$
of $G$ is the size of a smallest resolving set for $G$. Deciding whether a
given graph $G$ has Metric Dimension at most $k$ for some integer $k$ is
well-known to be NP-complete. Many research has been done to understand the
complexity of this problem on restricted graph classes. In this paper, we
decompose a graph into its so called $\textit{extended biconnected components}$
and present an efficient algorithm for computing the metric dimension for a
class of graphs having a minimum resolving set with a bounded number of
vertices in every extended biconnected component. Further we show that the
decision problem METRIC DIMENSION remains NP-complete when the above limitation
is extended to usual biconnected components.
