How do forbidden induced subgraphs impact global phenomena in graphs?

禁止诱导子图如何影响图中的全局现象？

• 批准号: RGPIN-2017-06673
• 负责人: Seamone, Benjamin
• 金额: $ 2.91万
• 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: 2022
• 资助国家: 加拿大
• 起止时间: 2022-01-01 至 2023-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=757438
• 关键词: How; do; forbidden; induced; subgraphs

My research program is in theoretical and algorithmic graph theory. I seek to better understand structural properties of graphs through the lens of parameters motivated by real world problems.A “graph” (or “network”) is a set of objects (“vertices”) and a set of pairs of vertices (“edges”). In most applications, graphs are used to represent a system of connections or relationships. The Facebook graph has users as its vertices and two users are connected by an edge if they are friends. A highway system can be modelled by a graph whose vertices are cities and towns, and the edges are roads and highways. A telecommunications network can be modelled by a graph where vertices are towers and two are connected by an edge if they can communicate with one another. Many problems we would like to solve in a graph theoretic setting are computationally very difficult. For example, suppose a delivery truck wishes to visit every city in a region exactly once in a single trip, finishing where it started. The time it takes to determine if such a route even exists grows extremely rapidly as the number of cities to consider increases. Under what conditions can we efficiently solve such a problem? Even better, what conditions necessarily imply a positive solution? My research program focuses on finding subgraphs which, when forbidden from occurring in a graph as an induced subgraph, guarantee a positive answer to a question which is otherwise difficult to solve. In the long term, I aim to deeply understand the effect of forbidding induced subgraphs on the following three aspects of a graph (examples of real world motivations for each aspect given in parentheses):(1) the existence of long cycles (efficient or optimal routing in networks),(2) covering the graph with particular subgraphs, especially complete graphs (optimizing computer performance, food webs, analysis of real world complex networks), and(3) the chromatic number of the graph, especially as it relates to other graph parameters (scheduling problems, channel assignment in communication networks).Over the next five years, I intend to make progress in all three themes. I am interested in generalizing known closure concepts in H-free graphs, a project that is large enough in scope to warrant support from both PhD students and postdoctoral fellows. These concepts are vital tools for guaranteeing spanning cycles in H-free graphs. Recent progress on edge clique coverings suggests new avenues to be explored regarding this parameter, including solving one remaining case of an open problem on covering the edges of claw-free graphs with cliques. Finally, I will pursue an open conjecture related to chi-boundedness which, surprisingly, relates induced subgraphs, paths and cycles, and graph colouring. Preliminary evidence suggests improvements can be made on the best known results related to the conjecture, and opportunities exist for contributions from HQPs of all levels.

我的研究计划是理论和算法图理论。我试图通过由现实世界问题动机的参数镜头更好地理解图形的结构属性。“图”（或“网络”）是一组对象（“ vertices”）和一组对顶点（“边缘”）。在大多数应用程序中，图表用于表示连接系统或关系系统。 Facebook图的用户是其顶点，如果他们是朋友，则两个用户通过边缘连接。高速公路系统可以通过其顶点是城市和城镇的图表来建模，边缘是道路和高速公路。电信网络可以通过图形建模，如果顶点为塔，则可以通过边缘连接两个，如果它们可以相互通信。我们想在图理论设置中解决的许多问题在计算上非常困难。例如，假设一辆送货卡车希望在一次旅行中准确访问一个地区的每个城市，从而完成它的开始。随着考虑增加的城市数量，确定这种途径是否存在的时间非常迅速。在什么条件下，我们可以有效解决这样的问题？更好的是，哪些条件一定意味着一个积极的解决方案？我的研究计划的重点是寻找子图，当禁止在图中以作为诱导的子图而被禁止在图表中保证对问题的积极答案，我旨在深入了解诱发子图对以下三个方面的效果，以下三个方面的效果（长期以来为parentes中给出的每个方面的示例）（在括号中给出了两种范围）（cy eftife cyc cyc cyc cyc cyc cyc cyc）具有特定子图的图表，尤其是完整的图表（优化计算机性能，食物网，对现实世界复杂网络的分析），以及（3）图形的色度数，尤其是与其他图形参数相关的（调度问题，通信网络中的频道分配）。我有兴趣将已知的封闭概念概括在无H-FROPH上，该项目的范围足够大，可以保证博士生和博士后研究员的支持。这些概念是保证在无h-f-for图中跨越周期的重要工具。边缘集团覆盖物上的最新进度表明，有关此参数的新途径，包括解决了用群集覆盖无爪图边缘的剩下的一个开放问题的情况。最后，我将寻求与卡尼结合的开放猜想，令人惊讶的是，相关的诱导子图，路径和周期以及图形着色。初步证据表明，可以根据与协议有关的最著名结果进行改进，并为所有级别的HQP贡献而存在机会。