In the representation of graph $G$ as an Edge Intersection Graph (EPG) of paths on a grid, if every vertex of $G$ is represented by a path on the grid and the corresponding vertices are adjacent, then two paths share a grid edge. In a monotonic EPG representation, all paths on the grid are in ascending order in both rows and columns. In the (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 the 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 numbers of outer-plane graphs and show that $b^m(G)\leqslant 2$ holds for all outer-plane graphs $G$. In addition, we characterize the maximal outer planar graph and the cacti with (monotonic) bend numbers equal to 0, 1, and 2 in terms of forbidden induced subgraphs. As a by-product, we get a low-order polynomial-time algorithm for constructing (monotonic) EPG representations with the smallest possible number of inflections of maximal outer planar graphs and cacti.
