点可见图的识别与复杂性
Recognition and Complexity of Point Visibility Graphs
摘要：
点可见性图是由平面上的一组点导出的图，其中每个顶点对应于一个点，并且只要两个对应的点彼此可见，两个顶点就相邻，即它们之间的开段不包含集合中的任何其他点。研究了点可见图的识别问题：给定一个简单的无向图，判断它是否是平面上某个点集的可见图。我们表明，这个问题是完整的实的存在理论。因此，这个问题就像判定一个多项式不等式系统的实数解的存在性一样困难。证明涉及简单的子结构，迫使在所有实现的一些可见性图，这是适用于Mnëv和Richter-Gebert的代数普遍性建设共线性。这解决了一个长期悬而未决的问题，并为其他类别的可见性图的分析铺平了道路。此外，作为我们的建设之一的推论，我们表明，存在点的可见性图，不承认任何几何实现与点具有整数坐标。
英文摘要：
A point visibility graph is a graph induced by a set of points in the plane, where every vertex corresponds to a point, and two vertices are adjacent whenever the two corresponding points are visible from each other, that is, the open segment between them does not contain any other point of the set. We study the recognition problem for point visibility graphs: given a simple undirected graph, decide whether it is the visibility graph of some point set in the plane. We show that the problem is complete for the existential theory of the reals. Hence the problem is as hard as deciding the existence of a real solution to a system of polynomial inequalities. The proof involves simple substructures forcing collinearities in all realizations of some visibility graphs, which are applied to the algebraic universality constructions of Mnëv and Richter-Gebert. This solves a longstanding open question and paves the way for the analysis of other classes of visibility graphs. Furthermore, as a corollary of one of our construction, we show that there exist point visibility graphs that do not admit any geometric realization with points having integer coordinates.
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。
