Hypergraph

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Sample of hypergraph:

X = {v 1, v 2, v 3, v 4, v 5, v 6, v 7}

E = {e 1, e 2, e 3, e 4}

= {{v 1, v 2, v 3},{v 2, v 3},

{v 3, v 5, v 6},{v 4}}

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 $\mathcal{P}(X) \backslash \{\varnothing\}$ , where $\mathcal{P}(X)$ 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.

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.

1 Terminology
2 Isomorphism and equality
- 2.1 Examples
3 Symmetric hypergraphs
4 Transversals
5 Incidence matrix
6 Hypergraph colouring
7 Partitions
8 Theorems
9 Generalizations
10 See also
11 References

[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

$X = \lbrace x_n | n \in N \rbrace$

that is, the vertices are indexed by an index $n\in N$ , and the edge set is

$E = \lbrace e_i | i\in I, e_i \subseteq X \rbrace$

with the edges $e i$ indexed by an index $i\in I$ .

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

$H_A=\left(A, \lbrace e_i \cap A | e_i \cap A \neq \varnothing \rbrace \right)$

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

$\left(X, \lbrace e_i | i\in J \rbrace \right)$

Given a subset $A\subseteq X$ , the section hypergraph is the partial hypergraph

$H \times A = \left(A, \lbrace e_i | i\in I, e_i \subseteq A \subseteq X \rbrace \right)$

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

$X_n = \lbrace e_i | x_n \in e_i \rbrace$

When a notion of equality is properly defined, as done below, hypergaphs can be seen to be self-dual:

$\left(H^*\right)^* = H$

[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 $H \simeq G$ if there exists a bijection

$\phi:X \to Y$

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

$H \simeq G$ if and only if $H^* \simeq G^*$ .

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 $H \cong G$ . 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 $H\equiv G$ if the isomorphism $φ$ has

φ(x n) = y n

and

φ(e i) = f π(i)

Note that

$H\equiv G$ if and only if $H^* \cong G^*$

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:

$\left(H^*\right) ^* = H$

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,

$e_1 \cap e_4 \cap e_6 = \lbrace a\rbrace$

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

$f_1 \cap f_4 \cap f_6 = \varnothing$

In this example, $H$ and $G$ are equivalent, $H\equiv G$ , and the duals are strongly isomorphic: $H^*\cong G^*$ .

[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 k-hypergraph. A graph is just a 2-uniform hypergraph.

The 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 $T\subseteq X$ 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 and optimization.

[edit] Incidence matrix

Let $V = \{v_1, v_2, ~\ldots, ~ v_n\}$ and $E = \{e_1, e_2, ~ \ldots ~ e_m\}$ . Every hypergraph has an $n \times m$ incidence matrix $A = (a i j)$ where

$a_{ij} = \left\{ \begin{matrix} 1 & \mathrm{if} ~ v_i \in e_j \\ 0 & \mathrm{otherwise} \end{matrix} \right.$

The transpose $A t$ of the incidence matrix defines a hypergraph $H^* = (V^*,\ E^*)$ called the dual of $H$ , where $V *$ is an m-element set and $E *$ is an n-element set of subsets of $V *$ . For $v^*_j \in V^*$ and $e^*_i \in E^*, ~ v^*_j \in e^*_i$ if and only if $a i j = 1$ .

[edit] Hypergraph colouring

Hypergraph colouring is defined as follows. Let $H = (V, E)$ be a hypergraph such that $\Vert V\Vert = n$ . Then $C=\{c_1, c_2, \ldots, c_n\}$ is a proper colouring of $H$ if and only if, for all $e \in E, \vert e\vert > 1,$ there exists $v_i, v_j \in e$ such that $c_i \neq c_j$ .

[edit] Partitions

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

$(X_1, X_2,\cdots,X_K)$

of the vertex set $X$ such that the subhypergraph $H_{X_k}$ generated by $X k$ is transitive for each $1\le k \le K$ , and such that

$\sum_{k=1}^K r\left(H_{X_k} \right) = r(H)$

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

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

[edit] Theorems

Many theorems involving graphs also hold for hypergraphs. Ramsey's theorem is a typical example. 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 $b\in e_1$ and $e_1\in e_2$ , it is not true that $b\in e_2$ . 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

$\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right]$

[edit] See also

P system

[edit] References

^ E. Dauber, in Graph theory, ed. F. Harary, Addison Wesley, (1969) p. 172.

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.

Categories: Set families | Graph families | Combinatorics

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