  For a given graph $G$, the metric and edge metric dimensions of $G$,
$\dim(G)$ and ${\rm edim}(G)$, are the cardinalities of the smallest possible
subsets of vertices in $V(G)$ such that they uniquely identify the vertices and
the edges of $G$, respectively, by means of distances. It is already known that
metric and edge metric dimensions are not in general comparable. Infinite
families of graphs with pendant vertices in which the edge metric dimension is
smaller than the metric dimension are already known. In this article, we
construct a 2-connected graph $G$ such that $\dim(G)=a$ and ${\rm edim}(G)=b$
for every pair of integers $a,b$, where $4\le b<a$. For this we use
subdivisions of complete graphs, whose metric dimension is in some cases
smaller than the edge metric dimension. Along the way, we present an upper
bound for the metric and edge metric dimensions of subdivision graphs under
some special conditions.
