Detecting Induced Graph Patterns

图是由称为顶点的节点和这些节点之间称为边的链接组成的网络，这些节点表示某种关系，例如社交网络中连接到其他个体的个体。图是无处不在的，不仅在科学和工程中，而且在真实的生活中，因为是研究是否一个给定的图H出现在另一个给定的图G的模式，使G可以转换为H通过一系列的操作。例如，一个社交网络G是否可以被压缩成一个更小（更容易分析）的网络H，而不会破坏太多的信息？这个例子表明，作为模式出现的概念取决于将G转换为H时允许的操作。我们考虑以下四个基本图形操作：1。顶点删除2.边缘删除3.边缘收缩4.顶点删除从图中移除顶点（及其相邻边）。边删除从图形中删除边。边收缩将边（u，v）的顶点u和v从图中移除，并用一个新的顶点替换它们，该新的顶点恰好与u或v先前相邻的那些剩余顶点相邻。顶点分解是从图中删除顶点v，其中只有两个邻居u和w，然后包含边（u，w）。结合这四个图操作导致十个基本的图包含关系。例如，一个图H被称为图G的子图，如果H可以通过一系列的顶点删除、边删除和边收缩（以及顶点解散）从G获得，而H是G的诱导子图，如果我们允许顶点删除和边收缩，但不允许边删除。每个图包含关系对应于一个决策问题：在指定的包含关系下，图G是否包含某个图H？为了回答这个问题，我们必须设计一个所谓的算法，它可以被看作是一组指令，就像准备一顿饭的食谱，但目的是把它变成一个计算机程序来自动解决问题。一个关键的方面是运行时间，即，计算机解决问题所需的时间。然而，很有可能该问题福尔斯离散优化问题的范畴，对于该离散优化问题，没有合理的快速算法是已知的，并且对于该离散优化问题，这样的算法的存在甚至被认为是不可能的。理论计算机科学和离散数学最重要和最基本的成就之一是Robertson和Seymour的Graph Minor Project。他们提供了没有禁止子图的图的结构特征，并设计了一种在立方时间内解决任何问题H-Minor的算法;后一个问题是决定给定图是否包含某个固定图H（即，这不是输入的一部分）作为一个小的。他们的理论在计算机科学和数学领域产生了深刻的成果。他们理论的一个重要结论是，任何允许边缘删除的包容问题都可以有效地解决。我们的首要目标是发展一个类似于罗伯逊和西摩的理论，但关于诱导包容关系，即，当不允许边删除时，图形操作。由于对它们的非诱导对应物有用的技术不再适用，诱导包含关系的基本理论，类似于罗伯逊和西摩的图小项目，在很大程度上是不存在的。我们的研究计划旨在改变这一点。

A graph is a network consisting of nodes called vertices and links between those nodes called edges representing some relationship such as individuals that are connected to other individuals in a social network. Graphs are ubiquitous, not only in science and engineering but also in real life, as is the study of whether a given graph H appears as a pattern within another given graph G so that G can be transformed to H via a sequence of operations. For instance, can a social network G be compressed to a smaller (and easier to analyze) network H without destroying too much information? This example shows that the notion of appearing as a pattern depends upon the operations allowed when transforming G into H. We consider the following four elementary graph operations: 1. vertex deletions2. edge deletions3. edge contractions4. vertex dissolutions.A vertex deletion removes a vertex (and its adjacent edges) from the graph. An edge deletion removes an edge from the graph. An edge contraction removes the vertices u and v of the edge (u,v) from the graph and replaces them by a new vertex that is made adjacent to precisely those remaining vertices to which u or v was previously adjacent. A vertex dissolution is the removal of a vertex v from the graph with exactly two neighbours u and w followed by the inclusion of the edge (u,w). Combining these four graph operations leads to ten essential graph containment relations. For example, a graph H is called a minor of a graph G if H can be obtained from G by a sequence of vertex deletions, edge deletions and edge contractions (and so also vertex dissolutions), whereas H is an induced minor of G if we do allow vertex deletions and edge contractions but no edge deletions. Each graph containment relation corresponds to a decision problem: subject to the specified containment relation, does a graph G contain some graph H?In order to answer this question we must design a so-called algorithm, which can be seen as a set of instructions, like a recipe for preparing a meal, but with the purpose to turn it into a computer program to solve the problem automatically. A crucial aspect is the running time, i.e., the time it will take the computer to solve the problem. However, it may well be possible that the problem falls into the category of discrete optimization problems for which no reasonably fast algorithm is known, and for which the existence of such an algorithm is even considered to be unlikely. One of the most important and fundamental achievements of Theoretical Computer Science and Discrete Mathematics is Robertson and Seymour's Graph Minor Project. They have provided a structural characterization of graphs without a forbidden minor and have designed an algorithm that solves any problem H-Minor in cubic time; the latter problem is to decide whether a given graph contains some fixed graph H (i.e., that is not part of the input) as a minor. Their theory has led to deep results across Computer Science and Mathematics. An important consequence of their theory is that any containment problem allowing edge deletions can be efficiently solved. Our over-arching aim is to develop a theory, similar to that of Robertson and Seymour, but on induced containment relations, i.e., when edge deletions are not permitted graph operations. As techniques that are useful for their non-induced counterparts can no longer be applied, a basic theory for induced containment relations, similar to the Graph Minor Project of Robertson and Seymour, is largely absent. Our research proposal aims to change this.