Hypergraph | EPFL Graph Search

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 GraphSearch.

In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a directed hypergraph is a pair , where is a set of elements called nodes, vertices, points, or elements and is a set of pairs of subsets of . Each of these pairs is called an edge or hyperedge; the vertex subset is known as its tail or domain, and as its head or codomain. The order of a hypergraph is the number of vertices in . The size of the hypergraph is the number of edges in . The order of an edge in a directed hypergraph is : that is, the number of vertices in its tail followed by the number of vertices in its head. The definition above generalizes from a directed graph to a directed hypergraph by defining the head or tail of each edge as a set of vertices ( or ) rather than as a single vertex. A graph is then the special case where each of these sets contains only one element. Hence any standard graph theoretic concept that is independent of the edge orders will generalize to hypergraph theory. Under one definition, an undirected hypergraph is a directed hypergraph which has a symmetric edge set: If then . For notational simplicity one can remove the "duplicate" hyperedges since the modifier "undirected" is precisely informing us that they exist: If then where means implicitly in. While graph edges connect only 2 nodes, hyperedges connect an arbitrary number of nodes. However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a k-uniform hypergraph is a hypergraph such that all its hyperedges have size k. (In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. An undirected hypergraph is also called a set system or a family of sets drawn from the universal set. Hypergraphs can be viewed as incidence structures.

About this result

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.

Related publications (13)

Related people (6)

Related concepts (49)

Related courses (6)

Related lectures (69)

Bipartite graph

In the mathematical field of graph theory, a bipartite graph (or bigraph) is a graph whose vertices can be divided into two disjoint and independent sets and , that is, every edge connects a vertex in to one in . Vertex sets and are usually called the parts of the graph. Equivalently, a bipartite graph is a graph that does not contain any odd-length cycles. The two sets and may be thought of as a coloring of the graph with two colors: if one colors all nodes in blue, and all nodes in red, each edge has endpoints of differing colors, as is required in the graph coloring problem.

Hypergraph

Glossary of graph theory

This is a glossary of graph theory. Graph theory is the study of graphs, systems of nodes or vertices connected in pairs by lines or edges.

Space-Efficient Representations of Graphs

Jakab Tardos

With the increasing prevalence of massive datasets, it becomes important to design algorithmic techniques for dealing with scenarios where the input to be processed does not fit in the memory of a sin

EPFL2022

Spectral Hypergraph Sparsifiers of Nearly Linear Size

Mikhail Kapralov, Jakab Tardos

Graph sparsification has been studied extensively over the past two decades, culminating in spectral sparsifiers of optimal size (up to constant factors). Spectral hypergraph sparsification is a natur

IEEE COMPUTER SOC2022

Towards Tight Bounds for Spectral Sparsification of Hypergraphs

Mikhail Kapralov, Jakab Tardos

Cut and spectral sparsification of graphs have numerous applications, including e.g. speeding up algorithms for cuts and Laplacian solvers. These powerful notions have recently been extended to hyperg

ASSOC COMPUTING MACHINERY2021

A first course in statistical network analysis and applications.

Singular cohomology is defined by dualizing the singular chain complex for spaces. We will study its basic properties, see how it acquires a multiplicative structure and becomes a graded commutative a

The students learn the theory and practice of basic concepts and techniques in algorithms. The course covers mathematical induction, techniques for analyzing algorithms, elementary data structures, ma

Explores tactical configurations, covering the minimal size of subsets needed to cover element sets and the concept of K-sets and base points.

Explores set coverings and uniformity theorems in hypergraphs, revealing insights into their structures.

Explores AI applications in chemistry, hypergraphs, single-step ritual synthesis, and retro-synthetic models evaluation.