Monotonic Representations of Outerplanar Graphs as Edge Intersection Graphs of Paths on a Grid

In a representation of a graph $G$ as an edge intersection graph of paths on a grid (EPG) every vertex of $G$ is represented by a path on a grid and two paths share a grid edge iff the corresponding vertices are adjacent. In a monotonic EPG representation every path on the grid is ascending in both rows and columns. In a (monotonic) $B_k$-EPG representation every path on the grid has at most $k$ bends. The (monotonic) bend number $b(G)$ ($b^m(G)$) of a graph $G$ is the smallest natural number $k$ for which there exists a (monotonic) $B_k$-EPG representation of $G$. In this paper we deal with the monotonic bend number of outerplanar graphs and show that $b^m(G)\leqslant 2$ holds for every outerplanar graph $G$. Moreover, we characterize the maximal outerplanar graphs and the cacti with (monotonic) bend number equal to $0$, $1$ and $2$ in terms of forbidden induced subgraphs. As a byproduct we obtain low-degree polynomial time algorithms to construct (monotonic) EPG representations with the smallest possible number of bends for maximal outerplanar graphs and cacti.