On $k$-Bend and Monotonic $\ell$-Bend Edge Intersection Graphs of Paths on a Grid

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.