权益分类	功能权益	普通用户	{{item.name}}会员
{{category.name}}	{{benefitItem.name}}

Detecting Induced Graph Patterns

检测诱导图模式

基本信息

批准号：
EP/K025090/1
负责人：
Daniel Paulusma
金额：
$ 46.31万
依托单位：
Durham University
依托单位国家：
英国
项目类别：
Research Grant
财政年份：
2013
资助国家：
英国
起止时间：
2013 至无数据
项目状态：
已结题

来源：
https://gtr.ukri.org/projects?ref=EP%2FK025090%2F1
关键词：
Detecting Induced Graph Patterns

项目摘要

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.

图是一个由节点组成的网络，这些节点被称为顶点，这些节点之间的链接被称为边，代表一些关系，比如在社交网络中连接到其他个体的个体。图无处不在，不仅在科学和工程中，而且在现实生活中，就像研究一个给定的图H是否在另一个给定的图G中作为一个模式出现，以便G可以通过一系列操作转换为H一样。例如，一个社交网络G能否在不破坏太多信息的情况下被压缩到一个更小（也更容易分析）的网络H ？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。我们考虑以下四种基本图操作：顶点deletions2。deletions3边缘。contractions4边缘。顶点关系破裂。顶点删除从图中删除一个顶点（及其相邻边）。边删除从图中删除一条边。边收缩从图中移除边（u,v）的顶点u和v，并用一个新顶点代替它们，该顶点与u或v之前相邻的剩余顶点相邻。顶点分解是从恰好有两个邻居u和w的图中移除顶点v，然后包含边（u,w）。结合这四种图运算可以得到十个基本的图包含关系。例如，如果H可以通过顶点删除、边删除和边收缩（也包括顶点分解）的序列从G得到，则图H称为图G的次次，而如果我们允许顶点删除和边收缩但不允许边删除，则H是G的诱导次次。每个图的包含关系对应一个决策问题：在给定的包含关系下，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，很有可能这个问题属于离散优化问题的范畴，其中没有合理的快速算法是已知的，并且这种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的图小项目。他们提供了没有禁止小调的图的结构表征，并设计了一种算法，可以在三次时间内解决任何h小调问题；后一个问题是确定给定图是否包含某个固定图H（即，它不是输入的一部分）作为次要图。他们的理论在计算机科学和数学领域产生了深远的影响。他们的理论的一个重要结论是，任何允许边缘删除的遏制问题都可以有效地解决。我们的首要目标是发展一个理论，类似于Robertson和Seymour的理论，但在诱导遏制关系上，即，当边缘删除不允许图操作时。由于对非诱导的对等物有用的技术不再适用，类似于Robertson和Seymour的Graph Minor Project的诱导遏制关系的基本理论在很大程度上是缺失的。我们的研究计划旨在改变这种状况。