# Hypergraph

### From Wikipedia, the free encyclopedia

In mathematics, a **hypergraph** is a generalization of a graph, where edges can connect any number of vertices. Formally, a hypergraph *H* is a pair *H* = (*X*,*E*) where *X* is a set of elements, called *nodes* or *vertices*, and *E* is a set of non-empty subsets of *X* called *hyperedges* or *links*. Therefore, *E* is a subset of , where is the power set of *X*.

While graph edges are pairs of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. However, it is often useful 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 triples, and so on.

A hypergraph is also called a **set system** or a **family of sets** drawn from the **universal set** *X*. Hypergraphs can be viewed as incidence structures and vice versa. In particular, there is a Levi graph corresponding to every hypergraph, and *vice versa*.

Unlike graphs, hypergraphs are difficult to draw on paper, so they tend to be studied using the nomenclature of set theory rather than the more pictorial descriptions (like 'trees','forests' and 'cycles') of graph theory. Special cases include the clutter, where no edge appears as a subset of another edge; and the abstract simplicial complex, which contains all subsets of every edge.

The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.

## Contents |

## [edit] Terminology

Because hypergraph links can have any cardinality, there are multiple, distinct notions of the concept of a subgraph: *subhypergraphs*, *partial hypergraphs* and *section hypergraphs*.

Let *H* = (*X*,*E*) be the hypergraph consisting of vertices

that is, the vertices are indexed by an index , and the edge set is

with the edges *e*_{i} indexed by an index .

A **subhypergraph** is a hypergraph with some vertices removed. Formally, the subhypergraph *H*_{A} induced by a subset *A* of *X* is defined as

The **partial hypergraph** is a hypergraph with some edges removed. Given a subset of the index set *I*, the partial hypergraph generated by *J* is the hypergraph

Given a subset , the **section hypergraph** is the partial hypergraph

The **dual** *H* ^{*} of *H* is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by {*e*_{i}} and whose edges are given by {*X*_{n}} where

When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involution, i.e.,

A **host graph** *host* (*H*) for a connected hypergraph *H* is a connected graph with the same vertex set such that every hyperedge of *H* induces a connected subgraph in *host* (*H*). For a disconnected hypergraph the host graph is the graph union of the hosts of its connected components.

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.

## [edit] Isomorphism and equality

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 hypergraph *H* = (*X*,*E*) is **isomorphic** to a hypergraph *G* = (*Y*,*F*), written as if there exists a bijection

and a permutation π of *I* such that

- φ(
*e*_{i}) =*f*_{π(i)}

The bijection φ is then called the isomorphism of the graphs. Note that

- if and only if .

When the edges of a hypergraph are explicitly labeled, one has the additional notion of *strong isomorphism*. One says that *H* is **strongly isomorphic** to *G* if the permutation is the identity. One then writes . Note that all strongly isomorphic graphs are isomorphic, but not vice-versa.

When the vertices of a hypergraph are explicitly labeled, one has the notions of *equivalence*, and also of *equality*. One says that *H* is **equivalent** to *G*, and writes if the isomorphism φ has

- φ(
*x*_{n}) =*y*_{n}

and

- φ(
*e*_{i}) =*f*_{π(i)}

Note that

- if and only if

If, in addition, the permutation π is the identity, one says that *H* equals *G*, and writes *H* = *G*. Note that, with this definition of equality, graphs are self-dual:

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

### [edit] Examples

Consider the hypergraph *H* with edges

*H*= {*e*_{1}= {*a*,*b*},*e*_{2}= {*b*,*c*},*e*_{3}= {*c*,*d*},*e*_{4}= {*d*,*a*},*e*_{5}= {*b*,*d*},*e*_{6}= {*a*,*c*}}

and

*G*= {*f*_{1}= {α,β},*f*_{2}= {β,γ},*f*_{3}= {γ,δ},*f*_{4}= {δ,α},*f*_{5}= {α,γ},*f*_{6}= {β,δ}}

Then clearly *H* and *G* are isomorphic (with φ(*a*) = α, *etc.*), but they are not strongly isomorphic. So, for example, in *H*, vertex *a* meets edges 1, 4 and 6, so that,

In graph *G*, there does not exist any vertex that meets edges 1, 4 and 6:

In this example, *H* and *G* are equivalent, , and the duals are strongly isomorphic: .

## [edit] Symmetric hypergraphs

The **rank** *r*(*H*) of a hypergraph *H* 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

**. A graph is just a 2-uniform hypergraph.**

*k*-hypergraphThe degree *d(v)* of a vertex *v* is the number of edges that contain it. *H* is ** k-regular** if every vertex has degree

*k*.

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

Two vertices *x* and *y* of *H* are called **symmetric** if there exists an automorphism such that φ(*x*) = *y*. Two edges *e*_{i} and *e*_{j} are said to be **symmetric** if there exists an automorphism such that φ(*e*_{i}) = *e*_{j}.

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. If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply **transitive**.

Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity.

## [edit] Transversals

A **transversal** or **hitting set** of a hypergraph *H* = (*X*, *E*) is a set that has nonempty intersection with every edge. A transversal *T* is called *minimal* if no proper subset of *T* is a transversal. The *transversal hypergraph* of *H* is the hypergraph (*X*, *F*) whose edge set *F* consists of all minimal transversals of *H*. Computing the transversal hypergraph has applications in machine learning and other fields of computer science, as game theory, indexing of database, SAT problem, Data minning and optimization.

## [edit] Incidence matrix

Let and . Every hypergraph has an incidence matrix *A* = (*a*_{ij}) where

The transpose *A*^{t} of the incidence matrix defines a hypergraph called the **dual** of *H*, where *V* ^{*} is an *m*-element set and *E* ^{*} is an *n*-element set of subsets of *V* ^{*} . For and if and only if *a*_{ij} = 1.

## [edit] Hypergraph coloring

Hypergraph colouring is defined as follows. Let *H* = (*V*,*E*) be a hypergraph such that . Then is a proper colouring of *H* if and only if, for all there exists such that .

## [edit] Partitions

A partition theorem due to E. Dauber^{[1]} states that, for an edge-transitive hypergraph *H* = (*X*,*E*), there exists a partition

of the vertex set *X* such that the subhypergraph generated by *X*_{k} is transitive for each , and such that

where *r*(*H*) is the rank of *H*.

As a corollary, an edge-transitive hypergraph that is not vertex-transitive is bicolorable.

Graph partitioning (and in particular, hypergraph partitioning) has many applications to IC design.^{[2]}

## [edit] Theorems

Many theorems and concepts involving graphs also hold for hypergraphs. Ramsey's theorem and Line graph of a hypergraph are typical examples. Some methods for studying symmetries of graphs extend to hypergraphs.

Two prominent theorems are the Erdős–Ko–Rado theorem and the Kruskal–Katona theorem on uniform hypergraphs.

## [edit] Generalizations

One possible generalization of a hypergraph is to allow edges to point at other edges. There are two variations of this generalization. In one, the edges consist not only of a set of vertices, but may also contain subsets of vertices, *ad infinitum*. Set membership then provides an ordering, but the ordering is neither a partial order nor a preorder, since it is not transitive. The graph corresponding to the Levi graph of this generalization is a directed acyclic graph. Consider, for example, the generalized hypergraph whose vertex set is *V* = {*a*,*b*} and whose edges are *e*_{1} = {*a*,*b*} and *e*_{2} = {*a*,*e*_{1}}. Then, although and , it is not true that . However, the transitive closure of set membership for such hypergraphs does induce a partial order, and "flattens" the hypergraph into a partially ordered set.

Alternately, edges can be allowed to point at other edges, (irrespective of the requirement that the edges be ordered as directed, acyclic graphs). This allows graphs with edge-loops, which need not contain vertices at all. For example, consider the generalized hypergraph consisting of two edges *e*_{1} and *e*_{2}, and zero vertices, so that *e*_{1} = {*e*_{2}} and *e*_{2} = {*e*_{1}}. As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation. In particular, there is no transitive closure of set membership for such hypergraphs. Although such structures may seem strange at first, they can be readily understood by noting that the equivalent generalization of their Levi graph is no longer bipartite, but is rather just some general directed graph.

The generalized incidence matrix for such hypergraphs is, by definition, a square matrix, of a rank equal to the total number of vertices plus edges. Thus, for the above example, the incidence matrix is simply

## [edit] See also

## [edit] Notes

**^**E. Dauber, in*Graph theory*, ed. F. Harary, Addison Wesley, (1969) p. 172.**^**Karypis, G., Aggarwal, R., Kumar, V., and Shekhar, S. (March 1999). "Multilevel hypergraph partitioning: applications in VLSI domain".*IEEE Transactions on Very Large Scale Integration (VLSI) Systems***7**(1): pp. 69–79. doi:. http://ieeexplore.ieee.org/xpls/abs_all.jsp?arnumber=748202.

## [edit] References

- Claude Berge, Dijen Ray-Chaudhuri, "Hypergraph Seminar, Ohio State University 1972",
*Lecture Notes in Mathematics***411**Springer-Verlag *This article incorporates material from hypergraph on PlanetMath, which is licensed under the GFDL.*