  In an EPG-representation of a graph $G$ each vertex is represented by a path
in the rectangular grid, and $(v,w)$ is an edge in $G$ if and only if the paths
representing $v$ an $w$ share a grid-edge. Requiring paths representing edges
to be x-monotone or, even stronger, both x- and y-monotone gives rise to three
natural variants of EPG-representations, one where edges have no monotonicity
requirements and two with the aforementioned monotonicity requirements. The
focus of this paper is understanding how small a grid can be achieved for such
EPG-representations with respect to various graph parameters.
  We show that there are $m$-edge graphs that require a grid of area
$\Omega(m)$ in any variant of EPG-representations. Similarly there are
pathwidth-$k$ graphs that require height $\Omega(k)$ and area $\Omega(kn)$ in
any variant of EPG-representations. We prove a matching upper bound of $O(kn)$
area for all pathwidth-$k$ graphs in the strongest model, the one where edges
are required to be both x- and y-monotone. Thus in this strongest model, the
result implies, for example, $O(n)$, $O(n \log n)$ and $O(n^{3/2})$ area bounds
for bounded pathwidth graphs, bounded treewidth graphs and all classes of
graphs that exclude a fixed minor, respectively. For the model with no
restrictions on the monotonicity of the edges, stronger results can be achieved
for some graph classes, for example an $O(n)$ area bound for bounded treewidth
graphs and $O(n \log^2 n)$ bound for graphs of bounded genus.
