Structure and algorithms for graphs with forbidden induced subgraphs

图由一个有限集(顶点)和它们之间的连接(边)组成。许多现实世界的数据结构都符合这个框架，从机场(如果有直达航班，其中两个是连接的)到社交媒体账户(如果他们是社交媒体网络上的朋友，就连接了两个)。图表算法塑造了我们的日常体验。许多航空公司使用简单的启发式方法，通过为东向和西向航班提供不同的选择来增加机上电影的种类，因为大多数乘客每个方向都旅行一次。其他问题可能需要(接近)最优解，这使得启发式方法的使用不太可行。例如，一家公司可能希望优化乘客在机场看到的广告的种类。制作广告的成本很高，因此该公司想知道如何使用尽可能少的不同广告，以确保在由直达航班连接的两个机场没有乘客看到相同的广告。 这可以用图来表示：给定一个图，目标是为每个顶点分配一个自然数，使得对于每条边，它的两个顶点具有不同的数字，并且使用的最大数字尽可能小。这就是所谓的图着色问题，它在计算上很难处理。更广泛地说，图问题经常出现在为由连接描述的系统寻找有效解决方案的上下文中，但当没有对输入图做出任何假设时，这些问题往往是难以解决的。 提出的研究的一个中心问题是：关于输入图的哪些结构假设使得经典图问题(如着色、最大稳定集和最大团)是有效可解的？通常，结构假设被表示为禁止的局部模式(诱导子图)，其目标是推导出关于避免这些模式的图的足够数量的全局信息，以便快速处理它们。这就引出了更多的问题：我们能给出一个图类中所有图的结构吗？在这个图形类中，我们可以证明某些图形参数相互依赖，但不是一般的吗？我们能证明在这个类的所有非常大的图表中都有一个共同的模式吗？ 这些问题的答案符合描述如果一个大的结构不包含一个特定的小结构是什么样子的更大的背景。与这个问题相关的开创性结果已经在其他领域获得或公布，例如图未成年人和拟阵未成年人。该课程为HQP提供了适合行业和学术职业的背景，从而使HQP受益。虽然这项提议的主要影响将体现在图论和一般理论计算机科学领域知识的进步上，但它也有可能为实际用途提供工具，包括交通网络、社交媒体以及资源分配和分配。

Graphs consist of a finite set (vertices) and connections between them (edges). Many real-world data structures fit into this framework, from airports (two of which are connected if there is non-stop flight between them) to social media accounts (two of which are connected if they are friends on the social media network). Algorithms for graphs shape our everyday experiences. Many airlines use a simple heuristic to increase the variety of in-flight movies by having a different selection for eastbound and westbound flights, since most passengers travel once in each direction. Other problems might require (close to) optimum solutions, making the use of heuristics less feasible. As an example, a company might wish to optimize the variety of advertisements that passengers see at airports. Creating an ad is expensive, and thus the company would like to know how they can use as few different ads as possible to ensure that when no passenger sees the same ad at two airports connected by a non-stop flight. This can be stated in terms of graphs: Given a graph, the goal is to assign a natural number to every vertex such that for every edge, its two vertices have different numbers, and the highest number used is as small as possible. This is known as the Graph Colouring Problem, which is computationally intractable. More generally, graph problems arise often in the context of finding an efficient solution for a system described by connections, but they tend to be intractable when no assumptions are made about the input graph. A central question of the proposed research is the following: Which structural assumptions about input graphs make classic graph problems (such as colouring, maximum stable set, and maximum clique) efficiently solvable? Usually, structural assumptions are formulated as forbidden local patterns (induced subgraphs), and the goal is to deduce a sufficient amount of global information about graphs that avoid these patterns to process them quickly. This gives rise to further questions: Can we give a construction for all graphs in a graph class? Can we show that certain graph parameters depend on each other in this graph class, but not in general? Can we show that there is a common pattern that occurs in all very large graphs in the class? Answers to these questions fit into the larger context of describing what a large structure looks like if it does not contain a specified smaller structure. Seminal results related to this question have been obtained or announced in other areas, such as graph minors and matroid minors. This programme benefits HQP by providing a background that is suitable both for industry and academic careers. While the primary impact of this proposal will be felt in the advancement of knowledge in the fields of graph theory, and theoretical computer science in general, it also has the potential to supply tools for practical uses including transportation networks, social media, and resource allocation and distribution.