Formally, a hypergraph is a pair where is a set of elements called nodes or vertices, and is a set of non-empty subsets of called hyperedges or edges.

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, it is a collection of sets of size k .) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on.

In computational geometry, a hypergraph may sometimes be called a range space and then the hyperedges are called ranges.

This circuit diagram can be interpreted as a drawing of a hypergraph in which four vertices ( depicted as white rectangles and disks ) are connected by three hyperedges drawn as trees.

An order-4 Venn diagram, which can be interpreted as a subdivision drawing of a hypergraph with 15 vertices ( the 15 colored regions ) and 4 hyperedges ( the 4 ellipses ).

An order-n Venn diagram, for instance, may be viewed as a subdivision drawing of a hypergraph with n hyperedges ( the curves defining the diagram ) and 2 < sup > n </ sup > − 1 vertices ( represented by the regions into which these curves subdivide the plane ).

A bipartite graph may be used to model a hypergraph in which is the set of vertices of the hypergraph, is the set of hyperedges, and contains an edge from a hypergraph vertex to a hypergraph edge exactly when is one of the endpoints of.

As a special case of this correspondence between bipartite graphs and hypergraphs, any multigraph ( a graph in which there may be two or more edges between the same two vertices ) may be interpreted as a hypergraph in which some hyperedges have equal sets of endpoints, and represented by a bipartite graph that does not have multiple adjacencies and in which the vertices on one side of the bipartition all have degree two.

A linear hypergraph is a hypergraph with the property that every two hyperedges have at most one vertex in common.

A hypergraph is said to be uniform if all of its hyperedges have the same number of vertices as each other.

The cliques of size in the Erdős – Faber – Lovász conjecture may be interpreted as the hyperedges of an-uniform linear hypergraph that has the same vertices as the underlying graph.

In this language, the Erdős – Faber – Lovász conjecture states that, given any-uniform linear hypergraph with hyperedges, one may-color the vertices such that each hyperedge has one vertex of each color.

A simple hypergraph is a hypergraph in which at most one hyperedge connects any pair of vertices and there are no hyperedges of size at most one.

While graph edges are pairs of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes.

In one possible visual representation for hypergraphs, similar to the standard graph drawing style in which curves in the plane are used to depict graph edges, a hypergraph's vertices are depicted as points, disks, or boxes, and its hyperedges are depicted as trees that have the vertices as their leaves.

If the vertices are represented as points, the hyperedges may also be shown as smooth curves that connect sets of points, or as simple closed curves that enclose sets of points.

That is, the sets of vertices represented by the hyperedges form a Sperner family.

Most classes of CSPs that are known to be tractable are those where the hypergraph of constraints has bounded treewidth ( and there are no restrictions on the set of constraint relations ), or where the constraints have arbitrary form but there exist essentially non-unary polymorphisms of the set of constraint relations.

Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called subhypergraphs, partial hypergraphs and section hypergraphs.

The dual of is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by and whose edges are given by where

A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BG, and ( x < sub > 1 </ sub >, e < sub > 1 </ sub >) are connected with an edge if and only if vertex x < sub > 1 </ sub > is contained in edge e < sub > 1 </ sub > in H. Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above.

When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism.

When the vertices of a hypergraph are explicitly labeled, one has the notions of equivalence, and also of equality.

A hypergraph is said to be vertex-transitive ( or vertex-symmetric ) if all of its vertices are symmetric.

Similarly, a hypergraph is edge-transitive if all edges are symmetric.

Ramsey's theorem and Line graph of a hypergraph are typical examples.

Consider, for example, the generalized hypergraph whose vertex set is and whose edges are and.

In the UNL approach, information conveyed by natural language is represented, sentence by sentence, as a hypergraph composed of a set of directed binary labeled links ( referred to as relations ) between nodes or hypernodes ( the Universal Words, or simply UW ), which stand for concepts.

Given a subset of the index set, the partial hypergraph generated by is the hypergraph

The transpose of the incidence matrix defines a hypergraph called the dual of, where is an m-element set and is an n-element set of subsets of.

In another style of hypergraph visualization, the subdivision model of hypergraph drawing, the plane is subdivided into regions, each of which represents a single vertex of the hypergraph.

In contrast with the polynomial-time recognition of planar graphs, it is NP-complete to determine whether a hypergraph has a planar subdivision drawing, but the existence of a drawing of this type may be tested efficiently when the adjacency pattern of the regions is constrained to be a path, cycle, or tree.

In mathematics, a hypergraph is a generalization of a graph in which an edge can connect any number of vertices.

The difference between a set system and a hypergraph ( which is not well defined ) is in the questions being asked.

A hypergraph is a combinatorial structure that, like an undirected graph, has vertices and edges, but in which the edges may be arbitrary sets of vertices rather than having to have exactly two endpoints.

* Link in graph theory may mean either the same as link in geometry or " edge of a hypergraph "

( Since a family of sets may be called a hypergraph, and since every set in has size r, is a uniform hypergraph of rank r .)

The edges of a hypergraph may form an arbitrary family of sets, so the line graph of a hypergraph is the same as the intersection graph of the sets from the family.

In particular, there is a bipartite " incidence graph " or " Levi graph " corresponding to every hypergraph, and conversely, most, but not all, bipartite graphs can be regarded as incidence graphs of hypergraphs.

Let be the hypergraph consisting of vertices

A hypergraph is bipartite if and only if its vertices can be partitioned into two classes U and V in such a way that each hyperedge contains at least one vertex from both classes.

If all edges have the same cardinality k, the hypergraph is said to be uniform or k-uniform, or is called a k-hypergraph.

Let be a hypergraph such that.