Topological Minors, Connectivity, and Partitions

科学和工程中的许多重要系统，包括无线通信、复杂网络（如Internet）和神经科学，都有图形的自然表述。要分析这样的系统，重要的是要理解相关图类的结构，并将这些图分解为结构化的部分。这样的结构信息和分解可以用来为这些模型中形成的问题提供良好的解决方案。本研究计划探讨若干基本图类的结构与分解问题。该项目产生的研究材料将用于培训该研究领域的本科生、研究生和早期职业研究人员。利用结构和极值技术，研究者和合作者最近解决了一个长期存在的关于不包含拓扑K5的图的猜想，以及关于图和超图分区的几个猜想。本项目将此研究扩展到研究相关问题，包括关于非分离路径的猜想以及关于图和超图的明智划分的几个问题。

Many important systems in science and engineering, including wireless communications, complex networks such as the Internet, and neural science, have natural formulations in terms of graphs. To analyze such systems, it is important to understand the structure of the associated classes of graphs and to decompose those graphs into structured pieces. Such structural information and decomposition can then be used to provide good solutions to problems formulated in these models. This research project investigates problems concerning structure and decomposition of certain fundamental classes of graphs. Research materials arising from the project will be used to train undergraduate, graduate, and early-career researchers in this field of research. Using structural and extremal techniques, the investigator and collaborators have recently resolved a longstanding conjecture on graphs containing no topological K5 and several conjectures on partitions of graphs and hypergraphs. This project extends this research to study related problems, including conjectures concerning non-separating paths and several problems concerning judicious partitions of graphs and hypergraphs.