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.
翻译:以图形 $G$ 表示在网格( EPG) 上路径的边缘交叉图示中, 每一个顶点 $G$ 以网格上的一条路径代表$G$, 如果相应的顶点是相邻的, 两条路径以网格上的一条网点边缘为代表一个网格。 在单调 EPG 中, 网格上的每一条路径都在上升。 在( monocontic) $B_ k$- EPG 中, 每条网格( monocicial) 代表着每个外平面图的弯曲点最多为$k美元。 此外, 图表( mononocicial) 的弯曲号是最小的自然数 $GG$ 。 在本文中, 我们处理外平面图的单调弯曲号数为$B$G$G$G$, 显示每个外平面图的弯曲号为$G$。 此外, 我们用最大外平面图和cactical$ 最小平面图( monocial$) 代表一个硬度的硬度数字, 等为美元。