关于网格上路径的 $k$-折与单调 $\ell$-折边交图
On $k$-Bend and Monotonic $\ell$-Bend Edge Intersection Graphs of Paths on a Grid
AI总结：
本文研究网格路径 EPG 表示中一般 $k$-折图类与单调 $k$-折图类的关系，借助完全二部图证明多个 $k$ 值下真包含，并给出折数增加的首个界。
摘要：
如果图 $G$ 可以用网格上的路径来表示，使得 $G$ 的每个顶点对应网格上的一条路径，并且 $G$ 中两个顶点相邻当且仅当相应路径共享一条网格边，那么该图称为 EPG 图，这种表示称为 EPG 表示。$k$-折 EPG 表示是指每条路径至多有 $k$ 个折的 EPG 表示。所有具有 $k$-折 EPG 表示的图组成的类记为 $B_k$。$B_\ell^m$ 是所有具有单调 $\ell$-折 EPG 表示的图组成的类，即一种 $\ell$-折 EPG 表示，其中每条路径在列方向和行方向上都单调上升。对所有 $k$，都有平凡关系 $B^m_k\subseteq B_k$。此外，已知当 $k=1$ 时 $B^m_k\subsetneqq B_k$。通过研究完全二部图的 $B_k$ 成员资格和 $B^m_k$ 成员资格，我们证明了当 $k\in\{2,3,5\}$ 以及 $k\geqslant 7$ 时，该包含关系同样是真包含。特别地，我们推导了这种成员资格要求 $m$、$n$ 和 $k$ 必须满足的必要条件，其中 $m$ 和 $n$ 是二部图两个分区类中的顶点数。我们猜想当 $k\in\{4,6\}$ 时也有 $B_k^m\subsetneqq B_k$。此外，我们证明对所有 $k\geqslant 5$，都有 $B_k\not\subseteq B_{2k-9}^m$。这意味着限制路径形状可能导致 EPG 表示所需折数显著增加。此前尚不知道这种增加幅度的界。我们证明了 $B_1\subseteq B_3^m$，给出了此类结果中的首个结论。
英文摘要：
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.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
