  If a graph $G$ can be represented by means of paths on a grid, such that each
vertex of $G$ corresponds to one path on the grid and two vertices of $G$ are
adjacent if and only if the corresponding paths share a grid edge, then this
graph is called EPG and the representation is called EPG representation. A
$k$-bend EPG representation is an EPG representation in which each path has at
most $k$ bends. The class of all graphs that have a $k$-bend EPG representation
is denoted by $B_k$. $B_\ell^m$ is the class of all graphs that have a
monotonic $\ell$-bend EPG representation, i.e. an $\ell$-bend EPG
representation, where each path is ascending in both columns and rows.
  It is trivial that $B^m_k\subseteq B_k$ for all $k$. Moreover, it is known
that $B^m_k\subsetneqq B_k$, for $k=1$. By investigating the $B_k$-membership
and the $B^m_k$-membership of complete bipartite graphs we prove that the
inclusion is also proper for $k\in \{2,3,5\}$ and for $k\geqslant 7$. In
particular, we derive necessary conditions for this membership that have to be
fulfilled by $m$, $n$ and $k$, where $m$ and $n$ are the number of vertices on
the two partition classes of the bipartite graph. We conjecture that $B_{k}^{m}
\subsetneqq B_{k}$ holds also for $k\in \{4,6\}$.
  Furthermore, we show that $B_k \not\subseteq B_{2k-9}^m$ holds for all
$k\geqslant 5$. This implies that restricting the shape of the paths can lead
to a significant increase of the number of bends needed in an EPG
representation. So far no bounds on the amount of that increase were known. We
prove that $B_1 \subseteq B_3^m$ holds, providing the first result of this
kind.
