平面直线图的最大夹角
Maximizing Maximal Angles for Plane Straight-Line Graphs
摘要：
设$G=(S, E)$是有限点集$S\subset\R^2$上一般位置的平面直线图。$G$的顶点$p \in S$的入射角是以入射到$p$的边的圆形顺序连续出现的$G$的任意两个边之间的角度。一个平面直线图称为$ϕ$-开图，如果每个顶点的入射角至少为$ϕ$。在本文中，我们研究了以下类型的问题：什么是最大角度$ϕ$，使任何有限集$S\subset\R^2$的点在一般的位置，我们可以找到一个图从某一类图的$S$是$ϕ$-开放？特别是，我们考虑类的三角剖分，生成树，路径上的$S$，并在大多数情况下给出紧界。
英文摘要：
Let $G=(S, E)$ be a plane straight-line graph on a finite point set $S\subset\R^2$ in general position. The incident angles of a vertex $p \in S$ of $G$ are the angles between any two edges of $G$ that appear consecutively in the circular order of the edges incident to $p$. A plane straight-line graph is called $ϕ$-open if each vertex has an incident angle of size at least $ϕ$. In this paper we study the following type of question: What is the maximum angle $ϕ$ such that for any finite set $S\subset\R^2$ of points in general position we can find a graph from a certain class of graphs on $S$ that is $ϕ$-open? In particular, we consider the classes of triangulations, spanning trees, and paths on $S$ and give tight bounds in most cases.
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