Geometric Representation of Graphs

这份提案概述了一个探索几何定义的图的性质的计划。例如，在交通、通信和传感器网络的研究中，就出现了这样的图表。所提出的研究重点是由几何对象之间的接触或可见性的概念定义的图。如果存在连接两个对象且不与另一个对象相交的线段，则这两个对象相互可见。如果他们的边界相交，而不是他们的内部相交，他们就是接触的。拟议研究的一个主要主题涉及同时表示，其中一组对象可以定义多个图。通过考虑对象之间在有限方向上的接触或可见性，我们获得了这些对象上有限数量的不同接触或可见性图。我的目标是了解这些可同时表示的图集的性质，并设计算法来找到这样的表示。*最常研究的同时表示的例子是寻找两个图中每个图的平面图的问题，其中每个顶点在两个图中都是相同的点(边是连接其端点的曲线)。这个问题的重要性超出了它的理论意义，因为它应用于一组公共顶点上不断变化的网络的可视分析。虽然点/曲线同时表示受到了极大的关注，但对可供选择的同时表示的研究要有限得多。我建议研究同时表示，使用几何对象，如线段、矩形和圆盘，作为可见性或接触决定邻接的顶点。例如，当可见性为垂直时，平面中的矩形定义一个图形；当可见性为水平时，平面中的矩形定义另一个图形。什么样的图形对可以用这种方式来表示？给出两个图，如果存在这样的表示，找到它的复杂性是什么？这种表示的存在是否有助于解决仅限于这些图的问题？*可以使用可见性隐式表示的图的类型根据对象的类型和可见性的概念而变化。这项拟议的研究的一部分考虑了允许一定程度的“X射线视觉”的可见性表示，即，如果存在连接两个对象的线段且该线段至多与k个其他对象相交，则两个对象是相互k-可见的。例如，当考虑可以穿透有限数量的墙的传感器网络时，就会出现这样的图表。该模型增加了可表示的图集，超出了传统的0-可见性表示，但受表示的几何形状的限制。拟议研究的一个目标是了解允许k-可见性在多大程度上影响可表示图的类，并将这类图与其他众所周知的图类联系起来。

This proposal outlines a plan to explore properties of graphs that are defined geometrically. Such graphs arise, for example, in the study of transportation, communication, and sensor networks. The focus of the proposed research is on graphs defined by some notion of contact or visibility between geometric objects. Two objects are mutually visible if there exists a line segment connecting them that does not intersect another object. They are in contact if their boundaries, but not their interiors, intersect. A main theme of the proposed research concerns simultaneous representation, where one set of objects can define more than one graph. By considering contact or visibility between objects in a limited number of directions, we obtain a limited number of different contact or visibility graphs on those objects. My goal is to understand properties of these sets of simultaneously representable graphs and to design algorithms to find such representations.******The most commonly studied example of simultaneous representation is the problem of finding a planar drawing of each of two graphs where each vertex is the same point in both drawings (and edges are curves that connect their endpoints). The problem is important, beyond its theoretical interest, for its application to the visual analysis of a changing network on a common set of vertices. While point/curve simultaneous representation has received a great deal of attention, the study of alternative simultaneous representations has been much more limited. I propose to study simultaneous representation using geometric objects, such as segments, rectangles, and disks, as vertices where visibility or contact determines adjacency. For example, rectangles in the plane define one graph when visibility is vertical and another when visibility is horizontal. What pairs of graphs can be represented in this fashion? Given two graphs, what is the complexity of finding such a representation if it exists? Can the existence of such a representation aid in the solving of problems restricted to these graphs?******The kinds of graphs that can be represented implicitly using visibility changes depending on the type of object and on the notion of visibility. Part of this proposed research considers visibility representations that allow some amount of "X-ray vision", that is, two objects are mutually k-visible if there is a segment connecting them that intersects at most k other objects. Such graphs arise, for example, when considering networks of sensors that can penetrate a limited number of walls. The model increases the set of graphs that can be represented beyond traditional 0-visibility representations, but in a way that is limited by the geometry of the representation. An objective of the proposed research is to understand to what extent permitting k-visibility impacts the class of representable graphs and to relate this class to other well-known graph classes.**