Structure and algorithms for graphs with forbidden induced subgraphs

图由一个有限集合（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（如果它们之间有直飞航班，其中两个是连接的）到社交媒体账户（如果它们是社交媒体网络上的朋友，其中两个是连接的）。图形算法塑造了我们的日常体验。许多航空公司使用一种简单的启发式方法，通过在东行和西行航班上选择不同的电影来增加机上电影的种类，因为大多数乘客在每个方向上都旅行一次。其他问题可能需要（接近）最优解决方案，使得启发式的使用不太可行。例如，公司可能希望优化乘客在机场看到的各种广告。制作一个广告是昂贵的，因此该公司想知道他们如何使用尽可能少的不同广告，以确保没有乘客在两个由直飞航班连接的机场看到相同的广告。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，这样对于每条边，它的两个顶点都有不同的数字，并且使用的最高数字尽可能小。这就是众所周知的图着色问题，这是一个难以计算的问题。更一般地说，图问题经常出现在寻找由连接描述的系统的有效解决方案的上下文中，但是当没有对输入图做出假设时，它们往往是难以处理的。提出的研究的一个中心问题是：输入图的哪些结构假设可以有效地解决经典的图问题（如着色、最大稳定集和最大团）？通常，结构假设被表述为禁止的局部模式（诱导子图），目标是推断出关于图的足够数量的全局信息，以避免这些模式以快速处理它们。这就产生了进一步的问题：我们能否给出一个图类中所有图的构造？我们能否证明在这个图类中某些图参数是相互依赖的，而不是一般的？我们能证明在这门课的所有大图表中都有一个共同的模式吗？这些问题的答案适用于描述大型结构在不包含特定较小结构的情况下的样子。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，例如图的次要和矩阵的次要。该课程为HQP提供了适合工业和学术生涯的背景。虽然这一建议的主要影响将体现在图论和理论计算机科学领域的知识进步上，但它也有可能为实际应用提供工具，包括交通网络、社交媒体、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.