[论文解读] Decomposition of Geometric Set Systems and Graphs

Dömötör Pálvölgyi|arXiv (Cornell University)|Jan 1, 2010

Computational Geometry and Mesh Generation参考文献 35被引用 7

一句话总结

该论文通过证明所有凸多边形都是覆盖可分解的，解决了Pach的猜想，表明当使用多边形的平移对平面进行足够高重覆盖时，可将其分解为k个不相交的覆盖。此外，论文还证明在三维空间中，没有凸多面体是覆盖可分解的，并为有界度数图的斜率数提供了紧致界，证明了立方图可用且仅用五种不同斜率进行绘制。

ABSTRACT

We study two decomposition problems in combinatorial geometry. The first part deals with the decomposition of multiple coverings of the plane. We say that a planar set is cover-decomposable if there is a constant m such that any m-fold covering of the plane with its translates is decomposable into two disjoint coverings of the whole plane. Pach conjectured that every convex set is cover-decomposable. We verify his conjecture for polygons. Moreover, if m is large enough, we prove that any m-fold covering can even be decomposed into k coverings. Then we show that the situation is exactly the opposite in 3 dimensions, for any polyhedron and any $m$ we construct an m-fold covering of the space that is not decomposable. We also give constructions that show that concave polygons are usually not cover-decomposable. We start the first part with a detailed survey of all results on the cover-decomposability of polygons. The second part investigates another geometric partition problem, related to planar representation of graphs. The slope number of a graph G is the smallest number s with the property that G has a straight-line drawing with edges of at most s distinct slopes and with no bends. We examine the slope number of bounded degree graphs. Our main results are that if the maximum degree is at least 5, then the slope number tends to infinity as the number of vertices grows but every graph with maximum degree at most 3 can be embedded with only five slopes. We also prove that such an embedding exists for the related notion called slope parameter. Finally, we study the planar slope number, defined only for planar graphs as the smallest number s with the property that the graph has a straight-line drawing in the plane without any crossings such that the edges are segments of only s distinct slopes. We show that the planar slope number of planar graphs with bounded degree is bounded.

研究动机与目标

• 解决关于平面中凸集覆盖可分解性的Pach猜想。
• 研究平面和空间中多重覆盖分解为不相交覆盖的问题。
• 确定最大度有界的图的斜率数和平面斜率数。
• 探讨超图移位链与覆盖可分解性之间的关系。
• 建立平面图和一般图的直线绘制所需斜率数的紧致界。

提出的方法

• 使用几何与组合论证，证明当m足够大时，任何以凸多边形进行的m重平面覆盖均可分解为k个不相交的覆盖。
• 在三维空间中构造显式反例，表明即使对于m重覆盖，也没有凸多面体是覆盖可分解的。
• 应用超图着色技术，特别是Property B和移位链构造，分析覆盖可分解性。
• 采用带斜率约束的图绘制技术，结合几何嵌入与斜率参数分析。
• 应用极值组合学与Lovász局部引理，界定平面图绘制中的斜率数量。
• 通过点集与凸集的几何转换，将斜率数问题转化为超图着色问题。

实验结果

研究问题

• RQ1 当覆盖重数m足够大时，每个凸多边形是否都能分解为k个不相交的覆盖？
• RQ2 三维空间中所有凸集是否都是覆盖可分解的？
• RQ3 最大度为4的图的斜率数是否无界？
• RQ4 最大度为d的每个平面图是否都能用O(d)种不同斜率绘制？
• RQ5 当k足够大时，k-一致超图的移位链是否具有Property B？

主要发现

• 当m足够大时，任何以凸多边形进行的m重平面覆盖均可分解为k个不相交的覆盖，其大小取决于k和多边形。
• 在三维空间中，对于任意凸多面体和任意m，均存在一个m重空间覆盖，无法分解为两个或更多个不相交的覆盖。
• 所有最大度至多为3的平面图均可仅用五种不同斜率绘制。
• 有界度数平面图的平面斜率数有界，且与顶点数无关。
• 最大度至少为5的图的斜率数随着顶点数增加而趋于无穷大。
• 最大度至多为3的图的斜率参数也有界，且此类图可仅用五种斜率嵌入。