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.

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 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 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 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

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.

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

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.

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

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.

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

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

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 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.

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.

Each hypergraph or set system can be regarded as an incidence

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

A factor graph is an example of a hypergraph, in that an arrow ( i. e., a factor node ) can connect more than one ( normal ) node.

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

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.

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.

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 .)

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.

However, the transitive closure of set membership for such hypergraphs does induce a partial order, and " flattens " the hypergraph into a partially ordered set.

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

A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge.

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.

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.