The set of automorphisms of a hypergraph H (= ( X, E )) is a group under composition, called the automorphism group of the hypergraph and written Aut ( H ).

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.

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

A hypergraph is also called a set system or a family of sets drawn from the universal set X.

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

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.

In computer science, a graph is an abstract data type that is meant to implement the graph and hypergraph concepts from mathematics.

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 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.

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.

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 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.

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.

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

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

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.

And, the hypergraph dual of vertex coloring is edge coloring.

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.

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

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

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

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

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.

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

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 ).

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 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 ).

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.

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.