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.

Computing the transversal hypergraph has applications in combinatorial optimization, in game theory, and in several fields of computer science such as machine learning, indexing of databases, the satisfiability problem, data mining, and computer program optimization.

As there is a hypergraph for every Levi graph, and vice-versa, the incidence matrix of an incidence structure describes a hypergraph.

Conversely, every incidence structure can be viewed as a hypergraph.

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

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.

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 >.

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 primal graph is sometimes also known as the Gaifman graph of the hypergraph.

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.

A graph is just a 2-uniform hypergraph.

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.

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 transversal ( or " hitting set ") of a hypergraph H = ( X, E ) is a set that has nonempty intersection with every edge.

Every hypergraph has an incidence matrix where

Graph partitioning ( and in particular, hypergraph partitioning ) has many applications to IC design and parallel computing.

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.

( 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 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.

In the graph coloring formulation of the Erdős – Faber – Lovász conjecture, it is safe to remove vertices that belong to a single clique, as their coloring presents no difficulty ; once this is done, the hypergraph that has a vertex for each clique, and a hyperedge for each graph vertex, forms a simple hypergraph.

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.

Let be the hypergraph consisting of vertices

A subhypergraph is a hypergraph with some vertices removed.

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 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 automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices.

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

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 ).

For example, consider the generalized hypergraph consisting of two edges and, and zero vertices, so that and.

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.

The partial hypergraph is a hypergraph with some edges removed.

Consider the hypergraph with edges

The rank of a hypergraph is the maximum cardinality of any of the edges in the hypergraph.

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

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

One possible generalization of a hypergraph is to allow edges to point at other edges.

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

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.