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.

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

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.

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

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

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.

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.

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

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.

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.

Let be a hypergraph such that.

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 ).

The hyperedges of the hypergraph are represented by contiguous subsets of these regions, which may be indicated by coloring, by drawing outlines around them, or both.

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 ).

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.

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.

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.

( 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 dual of a uniform hypergraph is regular and vice-versa.

A connected graph G with the same vertex set as a connected hypergraph H is a host graph for H if every hyperedge of H induces a connected subgraph in G. For a disconnected hypergraph H, G is a host graph if there is a bijection between the connected components of G and of H, such that each connected component G < nowiki >'</ nowiki > of G is a host of the corresponding H < nowiki >'</ nowiki >.

A hypergraph is isomorphic to a hypergraph, written as if there exists a bijection

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

Equivalently, a Sperner family is an antichain in the inclusion lattice over the power set of E. A Sperner family is also sometimes called an independent system or, if viewed from the hypergraph perspective, a clutter.

The primal graph of a hypergraph is the graph with the same vertices of the hypergraph, and edges between all pairs of vertices contained in the same hyperedge.

The transversal hypergraph of H is the hypergraph ( X, F ) whose edge set F consists of all minimal transversals of H.

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

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.

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

By contrast, a hypergraph can have multiple vertices assigned to one edge ; thus, the general case describes a hypergraph.

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

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.

Thus, the Erdős – Faber – Lovász conjecture is equivalent to the statement that any simple hypergraph with vertices has chromatic index ( edge coloring number ) at most.

The intersection number of a graph is the minimum number of elements in a family of sets whose intersection graph is, or equivalently the minimum number of vertices in a hypergraph whose line graph is.

define the linear intersection number of a graph, similarly, to be the minimum number of vertices in a linear hypergraph whose line graph is.

In the framework of edge coloring simple hypergraphs, defines a number from a simple hypergraph as the number of hypergraph vertices that belong to a hyperedge of three or more vertices.