基于Schnyder实现器的平面图紧致可见性表示的改进
Improved Compact Visibility Representation of Planar Graph via Schnyder's Realizer
摘要：
设$G$是一个$n$ -结点平面图.在$G$的可见性表示中，$G$的每个节点由水平线段表示，使得表示$G$的任何两个相邻节点的线段彼此垂直可见。本文给出了$G$的最佳可见性表示.给定三角化$G$的规范排序，我们的算法以贪婪的方式递增地绘制图形。我们表明，从施奈德的实现三角$G$得到的三个规范的顺序之一产生的$G$的可见性表示不宽于$\frac{22n-40}{15}$。我们易于实现的O（n）时间算法绕过了复杂的子程序的四个连接组件和四块树所需的最好的以前已知的算法康德。我们的结果提供了一个否定的答案，康德的开放性问题，$\frac{3n-6}{2}$是否是一个最坏情况下的下限所需的宽度。此外，如果$G$没有度为3（分别为度为5）的内部节点，则我们对$G$的可见性表示不会宽于$\frac{4n-9}{3}$（分别为$\frac{4n-7}{3}$）。此外，如果$G$是四连通的，那么我们对$G$的可见性表示并不比$n-1$宽，这与康德和He的最著名的结果相匹配。作为一个副产品，我们得到了一个更简单的证明瓦格纳定理的推论实现者，由于Bonichon，Saëc和Mosbah。
英文摘要：
Let $G$ be an $n$-node planar graph. In a visibility representation of $G$, each node of $G$ is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of $G$ are vertically visible to each other. In the present paper we give the best known compact visibility representation of $G$. Given a canonical ordering of the triangulated $G$, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained from Schnyder's realizer for the triangulated $G$ yields a visibility representation of $G$ no wider than $\frac{22n-40}{15}$. Our easy-to-implement O(n)-time algorithm bypasses the complicated subroutines for four-connected components and four-block trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant's open question about whether $\frac{3n-6}{2}$ is a worst-case lower bound on the required width. Also, if $G$ has no degree-three (respectively, degree-five) internal node, then our visibility representation for $G$ is no wider than $\frac{4n-9}{3}$ (respectively, $\frac{4n-7}{3}$). Moreover, if $G$ is four-connected, then our visibility representation for $G$ is no wider than $n-1$, matching the best known result of Kant and He. As a by-product, we obtain a much simpler proof for a corollary of Wagner's Theorem on realizers, due to Bonichon, Saëc, and Mosbah.
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