**Are you an EPFL student looking for a semester project?**

Work with us on data science and visualisation projects, and deploy your project as an app on top of Graph Search.

Concept# Perfect graph

Summary

In graph theory, a perfect graph is a graph in which the chromatic number equals the size of the maximum clique, both in the graph itself and in every induced subgraph. In all graphs, the chromatic number is greater than or equal to the size of the maximum clique, but they can be far apart. A graph is perfect when these numbers are equal, and remain equal after the deletion of arbitrary subsets of vertices.
The perfect graphs include many important families of graphs and serve to unify results relating colorings and cliques in those families. For instance, in all perfect graphs, the graph coloring problem, maximum clique problem, and maximum independent set problem can all be solved in polynomial time, despite their greater complexity for non-perfect graphs. In addition, several important minimax theorems in combinatorics, including Dilworth's theorem and Mirsky's theorem on partially ordered sets, Kőnig's theorem on matchings, and the Erdős–Szekeres theorem on monotonic sequences, can be expressed in terms of the perfection of certain associated graphs.
The perfect graph theorem states that the complement graph of a perfect graph is also perfect. The strong perfect graph theorem characterizes the perfect graphs in terms of certain forbidden induced subgraphs, leading to a polynomial time algorithm for testing whether a graph is perfect.
A clique in an undirected graph is a subset of its vertices that are all adjacent to each other.
A graph coloring assigns a color to each vertex so that each two adjacent vertices have different colors.
In particular, the vertices of any clique must all be assigned different colors, so the chromatic number (the minimum number of colors in any coloring) is always greater than or equal to the clique number, the number of vertices in the maximum clique. The perfect graphs are defined as the graphs for which these two numbers are equal, not just in the graph itself, but in every induced subgraph obtained by deleting some of its vertices.

Official source

This page is automatically generated and may contain information that is not correct, complete, up-to-date, or relevant to your search query. The same applies to every other page on this website. Please make sure to verify the information with EPFL's official sources.

Ontological neighbourhood

:

Related courses (11)

Related lectures (32)

Related publications (96)

Related people (22)

Related units (4)

Related concepts (60)

PHYS-512: Statistical physics of computation

This course covers the statistical physics approach to computer science problems ranging from graph theory and constraint satisfaction to inference and machine learning. In particular the replica and

MATH-602: Inference on graphs

The class covers topics related to statistical inference and algorithms on graphs: basic random graphs concepts, thresholds, subgraph containment (planted clique), connectivity, broadcasting on trees,

EE-548: Audio engineering

This lecture is oriented towards the study of audio engineering, with a special focus on room acoustics applications. The learning outcomes will be the techniques for microphones and loudspeaker desig

Fixed Points in Graph Theory

Focuses on fixed points in graph theory and their implications in algorithms and analysis.

Graphical Models: Joint Probability Distribution

Covers the concept of graphical models and joint probability distributions.

Statistical Physics of Clusters

Explores the statistical physics of clusters, focusing on complexity and equilibrium behavior.

Clique problem

In computer science, the clique problem is the computational problem of finding cliques (subsets of vertices, all adjacent to each other, also called complete subgraphs) in a graph. It has several different formulations depending on which cliques, and what information about the cliques, should be found. Common formulations of the clique problem include finding a maximum clique (a clique with the largest possible number of vertices), finding a maximum weight clique in a weighted graph, listing all maximal cliques (cliques that cannot be enlarged), and solving the decision problem of testing whether a graph contains a clique larger than a given size.

Interval graph

In graph theory, an interval graph is an undirected graph formed from a set of intervals on the real line, with a vertex for each interval and an edge between vertices whose intervals intersect. It is the intersection graph of the intervals. Interval graphs are chordal graphs and perfect graphs. They can be recognized in linear time, and an optimal graph coloring or maximum clique in these graphs can be found in linear time. The interval graphs include all proper interval graphs, graphs defined in the same way from a set of unit intervals.

Clique (graph theory)

In the mathematical area of graph theory, a clique (ˈkliːk or ˈklɪk) is a subset of vertices of an undirected graph such that every two distinct vertices in the clique are adjacent. That is, a clique of a graph is an induced subgraph of that is complete. Cliques are one of the basic concepts of graph theory and are used in many other mathematical problems and constructions on graphs. Cliques have also been studied in computer science: the task of finding whether there is a clique of a given size in a graph (the clique problem) is NP-complete, but despite this hardness result, many algorithms for finding cliques have been studied.

Philippe Schwaller, Junwu Chen

Graph neural networks (GNNs) have demonstrated promising performance across various chemistry-related tasks. However, conventional graphs only model the pairwise connectivity in molecules, failing to adequately represent higher order connections, such as m ...

Pascal Frossard, Chenglin Li, Mingxing Xu

Spectral-based graph neural networks (SGNNs) have been attracting increasing attention in graph representation learning. However, existing SGNNs are limited in implementing graph filters with rigid transforms and cannot adapt to signals residing on graphs ...

Pascal Frossard, Mireille El Gheche, Hermina Petric Maretic, Giovanni Chierchia

Graph comparison deals with identifying similarities and dissimilarities between graphs. A major obstacle is the unknown alignment of graphs, as well as the lack of accurate and inexpensive comparison metrics. In this work we introduce the filter graph dis ...