Graph searching - structural properties

图搜索-结构特性

• 批准号: RGPIN-2017-05065
• 负责人: Hahn, Gena
• 金额: $ 1.46万
• 依托单位: Université de Montréal
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2020
• 资助国家: 加拿大
• 起止时间: 2020-01-01 至 2021-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=711405
• 关键词: Graph; searching; structural; properties

Cops-and-robbers games have been studied since the 1980's, especially after the discovery of connections with various graph widths. Algorithmic and complexity questions seem to dominate the field because of applications in optimisation, among other things, but some basic questions remain unanswered: the cop-number of a graph, a characterisation of k-cop-win graphs, the length of an optimal game, etc. We wish to study these basic questions for the game as defined by Nowakowski and Winkler, and by Quillot, and to consider new models. Our graph theory group (two researchers, up to five students) has been studying the influence of loops on the number of cops needed (there are graphs that alternate between cop-win and not with the addition of loops one by one starting with a loopless graph). In the summer 2016 we have worked on a version of the game in which the cops have to catch the robber at a specified vertex (this is no longer a total knowledge game, the robber does not know the vertex). We wish to continue studying these variants where we have some preliminary results. Our algorithm to decide whether k cops can catch r robbers on a given (finite) graph is used in robotics for robot motion planning and we think that specifying a capture vertex and an algorithm to describe a strategy would also be useful to roboticists. Further, we would like to generalise by considering not only the number of cops needed to catch the robber, but also the cost of doing so. Perhaps having a few more cops would cost less? The main problem here is defining the cost. We have considered several possibilities and will continue in this direction. Since finite regular graphs are not cop-win unless complete, Cayley graphs have not been much considered. We think they are worth looking at again, especially in connection with some of the models (vaguely) described above. Last, we wish to study cop-win infinite graphs. We believe that understanding the infinite brings an understanding of the finite. Cops-and-robbers games on infinite graphs are different (there are many cop-win vertex transitive infinite graphs but only complete finite ones). A recent paper by Lehner suggests that we have been looking the problem backwards, overlooking the fact that the reverse of a well-order is a well-order for finite graphs but not for infinite ones. Thus any attempt at characterising infinite cop-win graphs through an ordering like that for finite ones is doomed to failure. This indicates that even for finite graphs we should reconsider our approach. We propose to do just that. A part of the study of infinite cop-win graphs are the implications to structural properties of graphs. Generalising a construction from our 2009 paper we have examples of universal countable graphs that are different from the unique (ultra)homogeneous countable graph (the random, or Rado, graph). We wish to continue studying such structures.

警匪片 自20世纪80年代以来，人们一直在研究游戏，特别是在 发现具有各种图形宽度的连接。数学和复杂性问题 似乎主导了该领域，因为在优化中的应用，除其他外，但一些基本的问题仍然没有答案：一个图的cop-number，k-cop-win图的表征，最优游戏的长度等。 我们希望研究Nowakowski定义的游戏的这些基本问题， Winkler和Quillot的研究，并考虑新的模型。我们的图论 小组（两名研究人员，最多五名学生）一直在研究 线圈对所需线圈数量的影响（有线圈绕组之间交替的图表 而不是从无环图开始一个接一个地添加循环）。 在2016年夏天 我们已经开发了一个游戏版本，在这个游戏中，警察必须抓住 强盗在指定的顶点（这不再是一个完全的知识游戏，强盗不知道顶点）。我们希望继续 研究这些变体我们有了一些初步的结果。我们的算法，以决定是否k警察可以抓住r劫匪在一个给定的 （有限）图在机器人学中用于机器人运动规划，我们认为指定捕获顶点和描述策略的算法对机器人专家也是有用的。 此外，我们想通过考虑不仅需要多少警察来抓住罪犯， 强盗，但这样做的成本。也许多几个警察 成本会更低？这里的主要问题是确定成本。我们已经考虑了几个 可能性，并将继续朝着这个方向发展。 由于有限正则图只有在完备的情况下才是cop-win的，因此Cayley图并没有得到太多的考虑。我们认为他们 值得再看一遍，特别是与一些模型（模糊）描述 以上 最后，我们希望研究cop-win无限图。我们相信，理解无限带来了对有限的理解。无限图上的Cops-and-Robbers博弈是不同的（有许多Cops-win顶点传递无限图，但只有完全有限图）。一 Lehner最近的一篇论文表明，我们一直在向后看这个问题，忽略了一个事实，即良序的逆对于有限图是良序，但对于无限图不是良序。因此，任何试图 通过类似于有限图的排序来刻画无限cop-win图注定会 失败这表明，即使对于 我们应该重新考虑我们的方法。我们建议这样做。 无限cop-win图的研究的一部分是图的结构性质的影响。一般化a 从我们2009年的论文中，我们有通用可数图的例子， 不同于唯一（超）齐次可数图（随机或Rado图）。 我们希望继续研究这种结构。