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创建页面，内容为“File:Hypergraph-wikipedia.svg|frame|An example of a hypergraph, with <math>X = \{v_1, v_2, v_3, v_4, v_5, v_6, v_7\}</math> and <math>E = \{e_1,e_2,e_3,e_4\} = </m…”
[[File:Hypergraph-wikipedia.svg|frame|An example of a hypergraph, with
<math>X = \{v_1, v_2, v_3, v_4, v_5, v_6, v_7\}</math> and
<math>E = \{e_1,e_2,e_3,e_4\} = </math>
<math>\{\{v_1, v_2, v_3\},</math>
<math>\{v_2,v_3\},</math>
<math>\{v_3,v_5,v_6\},</math>
<math>\{v_4\}\}</math>.
This hypergraph has order 7 and size 4. Here, edges do not just connect two vertices but several, and are represented by colors.]]
[[File:PAOH representation of the hypergraph.png|alt=PAOH visualization of a hypergraph|thumb|Alternative representation of the hypergraph reported in the figure above, called PAOH.<ref name="paoh">{{Cite journal|last=Valdivia|first=Paola|last2=Buono|first2=Paolo|last3=Plaisant|first3=Catherine|last4=Dufournaud|first4=Nicole|last5=Fekete|first5=Jean-Daniel|date=2020|title=Analyzing Dynamic Hypergraphs with Parallel Aggregated Ordered Hypergraph Visualization|journal=IEEE Transactions on Visualization and Computer Graphics|publisher=IEEE|volume=26|pages=12|doi=10.1109/TVCG.2019.2933196|pmid=31398121|issn=1077-2626|eissn=1941-0506}}</ref> Edges are vertical lines connecting vertices. Vertices are aligned on the left. The legend on the right shows the names of the edges.]]
{{Short description|Generalization of graph theory}}In [[mathematics]], a '''hypergraph''' is a generalization of a [[Graph (discrete mathematics)|graph]] in which an [[graph theory|edge]] can join any number of [[vertex (graph theory)|vertices]]. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a hypergraph <math>H</math> is a pair <math>H = (X,E)</math> where <math>X</math> is a set of elements called ''nodes'' or ''vertices'', and <math>E</math> is a set of non-empty subsets of <math>X</math> called ''[[hyperedges]]'' or ''edges''. Therefore, <math>E</math> is a subset of <math>\mathcal{P}(X) \setminus\{\emptyset\}</math>, where <math>\mathcal{P}(X)</math> is the [[power set]] of <math>X</math>. The size of the vertex set is called the ''order of the hypergraph'', and the size of edges set is the ''size of the hypergraph''.
While graph edges are 2-element subsets of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. 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, one such hypergraph is a collection of sets, each such set a hyperedge connecting ''k'' nodes.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. A hypergraph is also called a ''set system'' or a ''[[family of sets]]'' drawn from the [[universal set]].
Hypergraphs can be viewed as [[incidence structure]]s. In particular, there is a bipartite "incidence graph" or "[[Levi graph]]" corresponding to every hypergraph, and conversely, most, but not all, [[bipartite graph]]s can be regarded as incidence graphs of hypergraphs.
Hypergraphs have many other names. In [[computational geometry]], a hypergraph may sometimes be called a '''range space''' and then the hyperedges are called ''ranges''.<ref>{{citation
| last1 = Haussler | first1 = David | author1-link = David Haussler
| last2 = Welzl | first2 = Emo | author2-link = Emo Welzl
| doi = 10.1007/BF02187876
| issue = 2
| journal = [[Discrete and Computational Geometry]]
| mr = 884223
| pages = 127–151
| title = ε-nets and simplex range queries
| volume = 2
| year = 1987| doi-access = free
}}.</ref>
In [[cooperative game]] theory, hypergraphs are called '''simple games''' (voting games); this notion is applied to solve problems in [[social choice theory]]. In some literature edges are referred to as ''hyperlinks'' or ''connectors''.<ref>Judea Pearl, in ''HEURISTICS Intelligent Search Strategies for Computer Problem Solving'', Addison Wesley (1984), p. 25.</ref>
Special kinds of hypergraphs include: [[#Symmetric hypergraphs|''k''-uniform ones]], as discussed briefly above; [[clutter (mathematics)|clutter]]s, where no edge appears as a subset of another edge; and [[abstract simplicial complex]]es, which contain all subsets of every edge.
The collection of hypergraphs is a [[Category (mathematics)|category]] with hypergraph [[homomorphism]]s as [[morphism]]s.
==Terminology==
==== Definitions ====
There are different types of hypergraphs such as:
* ''Empty hypergraph'': a hypergraph with no edges.
* ''Non-simple (or multiple) hypergraph'': a hypergraph allowing loops (hyperedges with a single vertex) or repeated edges, which means there can be two or more edges containing the same set of vertices.
* ''Simple hypergraph'': a hypergraph that contains no loops and no repeated edges.
* ''<math>k </math>-uniform hypergraph'': a hypergraph where each edge contains precisely <math>k</math> vertices.
* ''<math>d </math>-regular hypergraph'': a hypergraph where every vertex has degree <math>d </math>.
* ''Acyclic hypergraph'': a hypergraph that does not contain any cycles.
Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called ''subhypergraphs'', ''partial hypergraphs'' and ''section hypergraphs''.
Let <math>H=(X,E)</math> be the hypergraph consisting of vertices
:<math>X = \lbrace x_i | i \in I_v \rbrace,</math>
and having ''edge set''
:<math>E = \lbrace e_i | i\in I_e \land e_i \subseteq X \land e_i \neq \emptyset \rbrace,</math>
where <math>I_v</math> and <math>I_e</math> are the [[index set]]s of the vertices and edges respectively.
A ''subhypergraph'' is a hypergraph with some vertices removed. Formally, the subhypergraph <math>H_A</math> induced by <math>A \subseteq X </math> is defined as
:<math>H_A=\left(A, \lbrace e \cap A | e \in E \land
e \cap A \neq \emptyset \rbrace \right).</math>
An ''extension'' of a ''subhypergraph'' is a hypergraph where each
hyperedge of <math>H</math> which is partially contained in the subhypergraph <math>H_A</math> and is fully contained in the extension <math>Ex(H_A)</math>.
Formally
:<math>Ex(H_A) = (A \cup A', E' )</math> with <math>A' = \bigcup_{e \in E} e \setminus A</math> and <math>E' = \lbrace e \in E | e \subseteq (A \cup A') \rbrace</math>.
The ''partial hypergraph'' is a hypergraph with some edges removed. Given a subset <math>J \subset I_e</math> of the edge index set, the partial hypergraph generated by <math>J</math> is the hypergraph
:<math>\left(X, \lbrace e_i | i\in J \rbrace \right).</math>
Given a subset <math>A\subseteq X</math>, the ''section hypergraph'' is the partial hypergraph
:<math>H \times A = \left(A, \lbrace e_i |
i\in I_e \land e_i \subseteq A \rbrace \right).</math>
The '''dual''' <math>H^*</math> of <math>H</math> is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by <math>\lbrace e_i \rbrace</math> and whose edges are given by <math>\lbrace X_m \rbrace</math> where
:<math>X_m = \lbrace e_i | x_m \in e_i \rbrace. </math>
When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an [[involution (mathematics)|involution]], i.e.,
:<math>\left(H^*\right)^* = H.</math>
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'' [[induced subgraph|induces]] a connected subgraph in ''G''. For a disconnected hypergraph ''H'', ''G'' is a host graph if there is a bijection between the [[connected component (graph theory)|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 ''bipartite'' if and only if its vertices can be partitioned into two classes ''U'' and ''V'' in such a way that each hyperedge with cardinality at least 2 contains at least one vertex from both classes. Alternatively, such a hypergraph is said to have [[Property B]].
The '''2-section''' (or '''clique graph''', '''representing graph''', '''primal graph''', '''Gaifman 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.
==Bipartite graph model==
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. This bipartite graph is also called [[incidence graph]].
==Acyclicity==
In contrast with ordinary undirected graphs for which there is a single natural notion of [[cycle (graph theory)|cycles]] and [[Forest (graph theory)|acyclic graphs]], there are multiple natural non-equivalent definitions of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the special case of ordinary graphs.
A first definition of acyclicity for hypergraphs was given by [[Claude Berge]]:<ref>[[Claude Berge]], ''Graphs and Hypergraphs''</ref> a hypergraph is Berge-acyclic if its [[incidence graph]] (the [[bipartite graph]] defined above) is acyclic. This definition is very restrictive: for instance, if a hypergraph has some pair <math>v \neq v'</math> of vertices and some pair <math>f \neq f'</math> of hyperedges such that <math>v, v' \in f</math> and <math>v, v' \in f'</math>, then it is Berge-cyclic. Berge-cyclicity can obviously be tested in [[linear time]] by an exploration of the incidence graph.
We can define a weaker notion of hypergraph acyclicity,<ref>C. Beeri, [[Ronald Fagin|R. Fagin]], D. Maier, [[Mihalis Yannakakis|M. Yannakakis]], ''On the Desirability of Acyclic Database Schemes''</ref> later termed α-acyclicity. This notion of acyclicity is equivalent to the hypergraph being conformal (every clique of the primal graph is covered by some hyperedge) and its primal graph being [[chordal graph|chordal]]; it is also equivalent to reducibility to the empty graph through the GYO algorithm<ref>C. T. Yu and M. Z. Özsoyoğlu. ''[https://www.computer.org/csdl/proceedings/cmpsac/1979/9999/00/00762509.pdf An algorithm for tree-query membership of a distributed query]''. In Proc. IEEE COMPSAC, pages 306-312, 1979</ref><ref name="graham1979universal">M. H. Graham. ''On the universal relation''. Technical Report, University of Toronto, Toronto, Ontario, Canada, 1979</ref> (also known as Graham's algorithm), a [[confluence (abstract rewriting)|confluent]] iterative process which removes hyperedges using a generalized definition of [[ear (graph theory)|ears]]. In the domain of [[database theory]], it is known that a [[database schema]] enjoys certain desirable properties if its underlying hypergraph is α-acyclic.<ref>[[Serge Abiteboul|S. Abiteboul]], [[Richard B. Hull|R. B. Hull]], [[Victor Vianu|V. Vianu]], ''Foundations of Databases''</ref> Besides, α-acyclicity is also related to the expressiveness of the [[guarded fragment]] of [[first-order logic]].
We can test in [[linear time]] if a hypergraph is α-acyclic.<ref>[[Robert Tarjan|R. E. Tarjan]], [[Mihalis Yannakakis|M. Yannakakis]]. ''Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs''. SIAM J. on Computing, 13(3):566-579, 1984.</ref>
Note that α-acyclicity has the counter-intuitive property that adding hyperedges to an α-cyclic hypergraph may make it α-acyclic (for instance, adding a hyperedge containing all vertices of the hypergraph will always make it α-acyclic). Motivated in part by this perceived shortcoming, [[Ronald Fagin]]<ref name="fagin1983degrees">[[Ronald Fagin]], ''Degrees of Acyclicity for Hypergraphs and Relational Database Schemes''</ref> defined the stronger notions of β-acyclicity and γ-acyclicity. We can state β-acyclicity as the requirement that all subhypergraphs of the hypergraph are α-acyclic, which is equivalent<ref name="fagin1983degrees"/> to an earlier definition by Graham.<ref name="graham1979universal"/> The notion of γ-acyclicity is a more restrictive condition which is equivalent to several desirable properties of database schemas and is related to [[Bachman diagram]]s. Both β-acyclicity and γ-acyclicity can be tested in [[PTIME|polynomial time]].
Those four notions of acyclicity are comparable: Berge-acyclicity implies γ-acyclicity which implies β-acyclicity which implies α-acyclicity. However, none of the reverse implications hold, so those four notions are different.<ref name="fagin1983degrees" />
==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 <math>H=(X,E)</math> is ''isomorphic'' to a hypergraph <math>G=(Y,F)</math>, written as <math>H \simeq G</math> if there exists a [[bijection]]
:<math>\phi:X \to Y</math>
and a [[permutation]] <math>\pi</math> of <math>I</math> such that
:<math>\phi(e_i) = f_{\pi(i)}</math>
The bijection <math>\phi</math> is then called the [[isomorphism]] of the graphs. Note that
:<math>H \simeq G</math> if and only if <math>H^* \simeq G^*</math>.
When the edges of a hypergraph are explicitly labeled, one has the additional notion of ''strong isomorphism''. One says that <math>H</math> is ''strongly isomorphic'' to <math>G</math> if the permutation is the identity. One then writes <math>H \cong G</math>. 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 <math>H</math> is ''equivalent'' to <math>G</math>, and writes <math>H\equiv G</math> if the isomorphism <math>\phi</math> has
:<math>\phi(x_n) = y_n</math>
and
:<math>\phi(e_i) = f_{\pi(i)}</math>
Note that
:<math>H\equiv G</math> if and only if <math>H^* \cong G^*</math>
If, in addition, the permutation <math>\pi</math> is the identity, one says that <math>H</math> equals <math>G</math>, and writes <math>H=G</math>. Note that, with this definition of equality, graphs are self-dual:
:<math>\left(H^*\right) ^* = H</math>
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 (mathematics)|group]] under composition, called the [[automorphism group]] of the hypergraph and written Aut(''H'').
===Examples===
Consider the hypergraph <math>H</math> with edges
:<math>H = \lbrace
e_1 = \lbrace a,b \rbrace,
e_2 = \lbrace b,c \rbrace,
e_3 = \lbrace c,d \rbrace,
e_4 = \lbrace d,a \rbrace,
e_5 = \lbrace b,d \rbrace,
e_6 = \lbrace a,c \rbrace
\rbrace</math>
and
:<math>G = \lbrace
f_1 = \lbrace \alpha,\beta \rbrace,
f_2 = \lbrace \beta,\gamma \rbrace,
f_3 = \lbrace \gamma,\delta \rbrace,
f_4 = \lbrace \delta,\alpha \rbrace,
f_5 = \lbrace \alpha,\gamma \rbrace,
f_6 = \lbrace \beta,\delta \rbrace
\rbrace</math>
Then clearly <math>H</math> and <math>G</math> are isomorphic (with <math>\phi(a)=\alpha</math>, ''etc.''), but they are not strongly isomorphic. So, for example, in <math>H</math>, vertex <math>a</math> meets edges 1, 4 and 6, so that,
:<math>e_1 \cap e_4 \cap e_6 = \lbrace a\rbrace</math>
In graph <math>G</math>, there does not exist any vertex that meets edges 1, 4 and 6:
:<math>f_1 \cap f_4 \cap f_6 = \varnothing</math>
In this example, <math>H</math> and <math>G</math> are equivalent, <math>H\equiv G</math>, and the duals are strongly isomorphic: <math>H^*\cong G^*</math>.
==Symmetric hypergraphs==
The ''{{visible anchor|rank}}'' <math>r(H)</math> of a hypergraph <math>H</math> 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 <math>\phi(x)=y</math>. Two edges <math>e_i</math> and <math>e_j</math> are said to be ''symmetric'' if there exists an automorphism such that <math>\phi(e_i)=e_j</math>.
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.
==Transversals==
A ''[[Transversal (combinatorics)|transversal]]'' (or "[[hitting set]]") of a hypergraph ''H'' = (''X'', ''E'') is a set <math>T\subseteq X</math> that has nonempty [[intersection (set theory)|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 [[combinatorial optimization]], in [[game theory]], and in several fields of [[computer science]] such as [[machine learning]], [[Index (database)|indexing of database]]s, the [[Boolean satisfiability problem|satisfiability problem]], [[data mining]], and computer [[program optimization]].
==Incidence matrix==
Let <math>V = \{v_1, v_2, ~\ldots, ~ v_n\}</math> and <math>E = \{e_1, e_2, ~ \ldots ~ e_m\}</math>. Every hypergraph has an <math>n \times m</math> [[incidence matrix]] <math>A = (a_{ij})</math> where
:<math>a_{ij} = \left\{ \begin{matrix} 1 & \mathrm{if} ~ v_i \in e_j \\ 0 & \mathrm{otherwise}. \end{matrix} \right.</math>
The [[transpose]] <math>A^t</math> of the [[incidence (geometry)|incidence]] matrix defines a hypergraph <math>H^* = (V^*,\ E^*)</math> called the '''dual''' of <math>H</math>, where <math>V^*</math> is an ''m''-element set and <math>E^*</math> is an ''n''-element set of subsets of <math>V^*</math>. For <math>v^*_j \in V^*</math> and <math>e^*_i \in E^*, ~ v^*_j \in e^*_i</math> [[if and only if]] <math>a_{ij} = 1</math>.
==Hypergraph coloring==
Classic hypergraph coloring is assigning one of the colors from set <math>\{1,2,3,...\lambda\}</math> to every vertex of a hypergraph in such a way that each hyperedge contains at least two vertices of distinct colors. In other words, there must be no monochromatic hyperedge with cardinality at least 2. In this sense it is a direct generalization of graph coloring. Minimum number of used distinct colors over all colorings is called the chromatic number of a hypergraph.
Hypergraphs for which there exists a coloring using up to ''k'' colors are referred to as ''k-colorable''. The 2-colorable hypergraphs are exactly the bipartite ones.
There are many generalizations of classic hypergraph coloring. One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. Some mixed hypergraphs are uncolorable for any number of colors. A general criterion for uncolorability is unknown. When a mixed hypergraph is colorable, then the minimum and maximum number of used colors are called the lower and upper chromatic numbers respectively. See http://spectrum.troy.edu/voloshin/mh.html for details.
==Partitions==
A partition theorem due to E. Dauber<ref>E. Dauber, in ''Graph theory'', ed. F. Harary, Addison Wesley, (1969) p. 172.</ref> states that, for an edge-transitive hypergraph <math>H=(X,E)</math>, there exists a [[partition of a set|partition]]
:<math>(X_1, X_2,\cdots,X_K)</math>
of the vertex set <math>X</math> such that the subhypergraph <math>H_{X_k}</math> generated by <math>X_k</math> is transitive for each <math>1\le k \le K</math>, and such that
:<math>\sum_{k=1}^K r\left(H_{X_k} \right) = r(H)</math>
where <math>r(H)</math> 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<ref>{{Citation |title=Multilevel hypergraph partitioning: applications in VLSI domain |author=Karypis, G., Aggarwal, R., Kumar, V., and Shekhar, S. |journal=IEEE Transactions on Very Large Scale Integration (VLSI) Systems |date=March 1999 |volume=7 |issue=1 |pages=69–79 |doi=10.1109/92.748202 |postscript=.|citeseerx=10.1.1.553.2367 }}</ref> and parallel computing.<ref>{{Citation |doi=10.1016/S0167-8191(00)00048-X |title=Graph partitioning models for parallel computing |author= Hendrickson, B., Kolda, T.G. |journal=Parallel Computing | year=2000 |volume=26 |issue=12 |pages=1519–1545 |postscript=.|url=https://digital.library.unt.edu/ark:/67531/metadc684945/ |type=Submitted manuscript }}</ref><ref>{{Cite conference |last1=Catalyurek |first1=U.V. |last2=Aykanat |first2=C. |title=A Hypergraph Model for Mapping Repeated Sparse Matrix-Vector Product Computations onto Multicomputers |conference=Proc. International Conference on Hi Performance Computing (HiPC'95) |year=1995}}</ref><ref>{{Citation |last1=Catalyurek |first1=U.V. |last2=Aykanat |first2=C. |title=Hypergraph-Partitioning Based Decomposition for Parallel Sparse-Matrix Vector Multiplication |journal=IEEE Transactions on Parallel and Distributed Systems |volume=10 |issue=7 |pages=673–693 |year=1999|doi=10.1109/71.780863 |postscript=. |citeseerx=10.1.1.67.2498 }}</ref> Efficient and Scalable [[Graph partition|hypergraph partitioning algorithms]] are also important for processing large scale hypergraphs in machine learning tasks.<ref name=hyperx>{{citation|last1=Huang|first1=Jin|last2=Zhang|first2=Rui|last3=Yu|first3=Jeffrey Xu|journal=Proceedings of the IEEE International Conference on Data Mining|title=Scalable Hypergraph Learning and Processing|year=2015}}</ref>
==Theorems==
Many [[theorem]]s 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.
==Hypergraph drawing==
[[File:CircuitoDosMallas.png|thumb|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.]]
Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs.
In one possible visual representation for hypergraphs, similar to the standard [[graph drawing]] style in which curves in the plane are used to depict graph edges, a hypergraph's vertices are depicted as points, disks, or boxes, and its hyperedges are depicted as trees that have the vertices as their leaves.<ref>{{citation
| last = Sander | first = G.
| contribution = Layout of directed hypergraphs with orthogonal hyperedges
| pages = 381–386
| publisher = Springer-Verlag
| series = [[Lecture Notes in Computer Science]]
| title = Proc. 11th International Symposium on Graph Drawing (GD 2003)
| contribution-url = http://gdea.informatik.uni-koeln.de/585/1/hypergraph.ps
| volume = 2912
| year = 2003| title-link = International Symposium on Graph Drawing
}}.</ref><ref>{{citation
| last1 = Eschbach | first1 = Thomas
| last2 = Günther | first2 = Wolfgang
| last3 = Becker | first3 = Bernd
| issue = 2
| journal = [[Journal of Graph Algorithms and Applications]]
| pages = 141–157
| title = Orthogonal hypergraph drawing for improved visibility
| url = http://jgaa.info/accepted/2006/EschbachGuentherBecker2006.10.2.pdf
| volume = 10
| year = 2006 | doi=10.7155/jgaa.00122}}.</ref> If the vertices are represented as points, the hyperedges may also be shown as smooth curves that connect sets of points, or as [[simple closed curve]]s that enclose sets of points.<ref>{{citation
| last = Mäkinen | first = Erkki
| doi = 10.1080/00207169008803875
| issue = 3
| journal = International Journal of Computer Mathematics
| pages = 177–185
| title = How to draw a hypergraph
| volume = 34
| year = 1990}}.</ref><ref>{{citation
| last1 = Bertault | first1 = François
| last2 = Eades | first2 = Peter | author2-link = Peter Eades
| contribution = Drawing hypergraphs in the subset standard
| doi = 10.1007/3-540-44541-2_15
| pages = 45–76
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| title = Proc. 8th International Symposium on Graph Drawing (GD 2000)
| volume = 1984
| year = 2001| title-link = International Symposium on Graph Drawing
| isbn = 978-3-540-41554-1
| doi-access = free
}}.</ref><ref>{{citation
| last1 = Naheed Anjum | first1 = Arafat
| last2 = Bressan | first2 = Stéphane
| contribution = Hypergraph Drawing by Force-Directed Placement
| doi = 10.1007/978-3-319-64471-4_31
| pages = 387–394
| publisher = Springer International Publishing
| series = Lecture Notes in Computer Science
| title = 28th International Conference on Database and Expert Systems Applications (DEXA 2017)
| volume = 10439
| year = 2017| isbn = 978-3-319-64470-7
}}.</ref>
[[File:Venn's four ellipse construction.svg|thumb|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).]]
In another style of hypergraph visualization, the subdivision model of hypergraph drawing,<ref>{{citation
| last1 = Kaufmann | first1 = Michael
| last2 = van Kreveld | first2 = Marc
| last3 = Speckmann | first3 = Bettina | author3-link = Bettina Speckmann
| contribution = Subdivision drawings of hypergraphs
| doi = 10.1007/978-3-642-00219-9_39
| pages = 396–407
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| title = Proc. 16th International Symposium on Graph Drawing (GD 2008)
| volume = 5417
| year = 2009| title-link = International Symposium on Graph Drawing
| isbn = 978-3-642-00218-2
| doi-access = free
}}.</ref> the plane is subdivided into regions, each of which represents a single vertex of the hypergraph. 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. 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). In contrast with the polynomial-time recognition of [[planar graph]]s, it is [[NP-complete]] to determine whether a hypergraph has a planar subdivision drawing,<ref>{{citation
| last1 = Johnson | first1 = David S. | author1-link = David S. Johnson
| last2 = Pollak | first2 = H. O.
| doi = 10.1002/jgt.3190110306
| issue = 3
| journal = Journal of Graph Theory
| pages = 309–325
| title = Hypergraph planarity and the complexity of drawing Venn diagrams
| volume = 11
| year = 2006}}.</ref> 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.<ref>{{citation
| last1 = Buchin | first1 = Kevin
| last2 = van Kreveld | first2 = Marc
| last3 = Meijer | first3 = Henk
| last4 = Speckmann | first4 = Bettina
| last5 = Verbeek | first5 = Kevin
| contribution = On planar supports for hypergraphs
| doi = 10.1007/978-3-642-11805-0_33
| pages = 345–356
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| title = Proc. 17th International Symposium on Graph Drawing (GD 2009)
| volume = 5849
| year = 2010| title-link = International Symposium on Graph Drawing
| isbn = 978-3-642-11804-3
| doi-access = free
}}.</ref>
An alternative representation of the hypergraph called PAOH<ref name="paoh" /> is shown in the figure on top of this article. Edges are vertical lines connecting vertices. Vertices are aligned on the left. The legend on the right shows the names of the edges. It has been designed for dynamic hypergraphs but can be used for simple hypergraphs as well.
==Hypergraph grammars==
{{main|Hypergraph grammar}}
By augmenting a class of hypergraphs with replacement rules, [[graph grammar]]s can be generalised to allow hyperedges.
==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, subsets of subsets of vertices and so on ''ad infinitum''. In essence, every edge is just an internal node of a tree or [[directed acyclic graph]], and vertices are the leaf nodes. A hypergraph is then just a collection of trees with common, shared nodes (that is, a given internal node or leaf may occur in several different trees). Conversely, every collection of trees can be understood as this generalized hypergraph. Since trees are widely used throughout [[computer science]] and many other branches of mathematics, one could say that hypergraphs appear naturally as well. So, for example, this generalization arises naturally as a model of [[term algebra]]; edges correspond to [[term (logic)|terms]] and vertices correspond to constants or variables.
For such a hypergraph, 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 <math>V= \{a,b\}</math> and whose edges are <math>e_1=\{a,b\}</math> and <math>e_2=\{a,e_1\}</math>. Then, although <math>b\in e_1</math> and <math>e_1\in e_2</math>, it is not true that <math>b\in e_2</math>. 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 <math>e_1</math> and <math>e_2</math>, and zero vertices, so that <math>e_1 = \{e_2\}</math> and <math>e_2 = \{e_1\}</math>. 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 graph|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
:<math>\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right].</math>
==Hypergraph learning==
Hypergraphs have been extensively used in [[machine learning]] tasks as the data model and classifier [[regularization (mathematics)]].<ref>{{citation| last1 = Zhou | first1 = Dengyong| last2 = Huang | first2 = Jiayuan | last3=Scholkopf | first3=Bernhard| issue = 2| journal = Advances in Neural Information Processing Systems| pages = 1601–1608| title = Learning with hypergraphs: clustering, classification, and embedding| year = 2006}}</ref> The applications include [[recommender system]] (communities as hyperedges),<ref>{{citation|last1=Tan | first1=Shulong | last2=Bu | first2=Jiajun | last3=Chen | first3=Chun | last4=Xu | first4=Bin | last5=Wang | first5=Can | last6=He | first6=Xiaofei|issue = 1| journal = ACM Transactions on Multimedia Computing, Communications, and Applications| title = Using rich social media information for music recommendation via hypergraph model| year = 2013|url=https://www.researchgate.net/publication/226075153| bibcode=2011smma.book..213T }}</ref> [[image retrieval]] (correlations as hyperedges),<ref>{{citation|last1=Liu | first1=Qingshan | last2=Huang | first2=Yuchi | last3=Metaxas | first3=Dimitris N. |issue = 10–11| journal = Pattern Recognition| title = Hypergraph with sampling for image retrieval| pages=2255–2262| year = 2013| doi=10.1016/j.patcog.2010.07.014 | volume=44}}</ref> and [[bioinformatics]] (biochemical interactions as hyperedges).<ref>{{citation|last1=Patro |first1=Rob | last2=Kingsoford | first2=Carl| issue = 10–11| journal = Bioinformatics| title = Predicting protein interactions via parsimonious network history inference| year = 2013| pages=237–246|doi=10.1093/bioinformatics/btt224 |pmid=23812989 |pmc=3694678 | volume=29}}</ref> Representative hypergraph learning techniques include hypergraph [[spectral clustering]] that extends the [[spectral graph theory]] with hypergraph Laplacian,<ref>{{citation|last1=Gao | first1=Tue | last2=Wang | first2=Meng | last3=Zha|first3=Zheng-Jun|last4=Shen|first4=Jialie|last5=Li|first5=Xuelong|last6=Wu|first6=Xindong|issue = 1| journal = IEEE Transactions on Image Processing| volume=22 | title = Visual-textual joint relevance learning for tag-based social image search| year = 2013| pages=363–376|url=http://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=2510&context=sis_research | doi=10.1109/tip.2012.2202676| pmid=22692911 | bibcode=2013ITIP...22..363Y }}</ref> and hypergraph [[semi-supervised learning]] that introduces extra hypergraph structural cost to restrict the learning results.<ref>{{citation|last1=Tian|first1=Ze|last2=Hwang|first2=TaeHyun|last3=Kuang|first3=Rui|issue = 21| journal = Bioinformatics| title = A hypergraph-based learning algorithm for classifying gene expression and arrayCGH data with prior knowledge| year = 2009| pages=2831–2838|doi=10.1093/bioinformatics/btp467|pmid=19648139| volume=25|doi-access=free}}</ref> For large scale hypergraphs, a distributed framework<ref name=hyperx /> built using [[Apache Spark]] is also available.
==See also==
{{Commons category|Hypergraphs}}
* [[Simplicial complex]]
* [[Combinatorial design]]
* [[Factor graph]]
* [[Greedoid]]
* [[Incidence structure]]
* [[Matroid]]
* [[Multigraph]]
* [[P system]]
* [[Sparse matrix-vector multiplication]]
*[[Matching in hypergraphs]]
==Notes==
{{Reflist}}
==References==
* Claude Berge, "Hypergraphs: Combinatorics of finite sets". North-Holland, 1989.
* Claude Berge, Dijen Ray-Chaudhuri, "Hypergraph Seminar, Ohio State University 1972", ''Lecture Notes in Mathematics'' '''411''' Springer-Verlag
* {{springer|title=Hypergraph|id=p/h048470}}
* Alain Bretto, "Hypergraph Theory: an Introduction", Springer, 2013.
* Vitaly I. Voloshin. "Coloring Mixed Hypergraphs: Theory, Algorithms and Applications". Fields Institute Monographs, American Mathematical Society, 2002.
* Vitaly I. Voloshin. "Introduction to Graph and Hypergraph Theory". [[Nova Science Publishers, Inc.]], 2009.
* {{PlanetMath attribution|urlname=hypergraph|title=hypergraph}}
==External links==
* [https://www.aviz.fr/paohvis PAOHVis]: open-source PAOHVis system for visualizing dynamic hypergraphs.
{{Graph representations}}
[[Category:Hypergraphs| ]]
[[de:Graph (Graphentheorie)#Hypergraph]]
<math>X = \{v_1, v_2, v_3, v_4, v_5, v_6, v_7\}</math> and
<math>E = \{e_1,e_2,e_3,e_4\} = </math>
<math>\{\{v_1, v_2, v_3\},</math>
<math>\{v_2,v_3\},</math>
<math>\{v_3,v_5,v_6\},</math>
<math>\{v_4\}\}</math>.
This hypergraph has order 7 and size 4. Here, edges do not just connect two vertices but several, and are represented by colors.]]
[[File:PAOH representation of the hypergraph.png|alt=PAOH visualization of a hypergraph|thumb|Alternative representation of the hypergraph reported in the figure above, called PAOH.<ref name="paoh">{{Cite journal|last=Valdivia|first=Paola|last2=Buono|first2=Paolo|last3=Plaisant|first3=Catherine|last4=Dufournaud|first4=Nicole|last5=Fekete|first5=Jean-Daniel|date=2020|title=Analyzing Dynamic Hypergraphs with Parallel Aggregated Ordered Hypergraph Visualization|journal=IEEE Transactions on Visualization and Computer Graphics|publisher=IEEE|volume=26|pages=12|doi=10.1109/TVCG.2019.2933196|pmid=31398121|issn=1077-2626|eissn=1941-0506}}</ref> Edges are vertical lines connecting vertices. Vertices are aligned on the left. The legend on the right shows the names of the edges.]]
{{Short description|Generalization of graph theory}}In [[mathematics]], a '''hypergraph''' is a generalization of a [[Graph (discrete mathematics)|graph]] in which an [[graph theory|edge]] can join any number of [[vertex (graph theory)|vertices]]. In contrast, in an ordinary graph, an edge connects exactly two vertices. Formally, a hypergraph <math>H</math> is a pair <math>H = (X,E)</math> where <math>X</math> is a set of elements called ''nodes'' or ''vertices'', and <math>E</math> is a set of non-empty subsets of <math>X</math> called ''[[hyperedges]]'' or ''edges''. Therefore, <math>E</math> is a subset of <math>\mathcal{P}(X) \setminus\{\emptyset\}</math>, where <math>\mathcal{P}(X)</math> is the [[power set]] of <math>X</math>. The size of the vertex set is called the ''order of the hypergraph'', and the size of edges set is the ''size of the hypergraph''.
While graph edges are 2-element subsets of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. 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, one such hypergraph is a collection of sets, each such set a hyperedge connecting ''k'' nodes.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. A hypergraph is also called a ''set system'' or a ''[[family of sets]]'' drawn from the [[universal set]].
Hypergraphs can be viewed as [[incidence structure]]s. In particular, there is a bipartite "incidence graph" or "[[Levi graph]]" corresponding to every hypergraph, and conversely, most, but not all, [[bipartite graph]]s can be regarded as incidence graphs of hypergraphs.
Hypergraphs have many other names. In [[computational geometry]], a hypergraph may sometimes be called a '''range space''' and then the hyperedges are called ''ranges''.<ref>{{citation
| last1 = Haussler | first1 = David | author1-link = David Haussler
| last2 = Welzl | first2 = Emo | author2-link = Emo Welzl
| doi = 10.1007/BF02187876
| issue = 2
| journal = [[Discrete and Computational Geometry]]
| mr = 884223
| pages = 127–151
| title = ε-nets and simplex range queries
| volume = 2
| year = 1987| doi-access = free
}}.</ref>
In [[cooperative game]] theory, hypergraphs are called '''simple games''' (voting games); this notion is applied to solve problems in [[social choice theory]]. In some literature edges are referred to as ''hyperlinks'' or ''connectors''.<ref>Judea Pearl, in ''HEURISTICS Intelligent Search Strategies for Computer Problem Solving'', Addison Wesley (1984), p. 25.</ref>
Special kinds of hypergraphs include: [[#Symmetric hypergraphs|''k''-uniform ones]], as discussed briefly above; [[clutter (mathematics)|clutter]]s, where no edge appears as a subset of another edge; and [[abstract simplicial complex]]es, which contain all subsets of every edge.
The collection of hypergraphs is a [[Category (mathematics)|category]] with hypergraph [[homomorphism]]s as [[morphism]]s.
==Terminology==
==== Definitions ====
There are different types of hypergraphs such as:
* ''Empty hypergraph'': a hypergraph with no edges.
* ''Non-simple (or multiple) hypergraph'': a hypergraph allowing loops (hyperedges with a single vertex) or repeated edges, which means there can be two or more edges containing the same set of vertices.
* ''Simple hypergraph'': a hypergraph that contains no loops and no repeated edges.
* ''<math>k </math>-uniform hypergraph'': a hypergraph where each edge contains precisely <math>k</math> vertices.
* ''<math>d </math>-regular hypergraph'': a hypergraph where every vertex has degree <math>d </math>.
* ''Acyclic hypergraph'': a hypergraph that does not contain any cycles.
Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called ''subhypergraphs'', ''partial hypergraphs'' and ''section hypergraphs''.
Let <math>H=(X,E)</math> be the hypergraph consisting of vertices
:<math>X = \lbrace x_i | i \in I_v \rbrace,</math>
and having ''edge set''
:<math>E = \lbrace e_i | i\in I_e \land e_i \subseteq X \land e_i \neq \emptyset \rbrace,</math>
where <math>I_v</math> and <math>I_e</math> are the [[index set]]s of the vertices and edges respectively.
A ''subhypergraph'' is a hypergraph with some vertices removed. Formally, the subhypergraph <math>H_A</math> induced by <math>A \subseteq X </math> is defined as
:<math>H_A=\left(A, \lbrace e \cap A | e \in E \land
e \cap A \neq \emptyset \rbrace \right).</math>
An ''extension'' of a ''subhypergraph'' is a hypergraph where each
hyperedge of <math>H</math> which is partially contained in the subhypergraph <math>H_A</math> and is fully contained in the extension <math>Ex(H_A)</math>.
Formally
:<math>Ex(H_A) = (A \cup A', E' )</math> with <math>A' = \bigcup_{e \in E} e \setminus A</math> and <math>E' = \lbrace e \in E | e \subseteq (A \cup A') \rbrace</math>.
The ''partial hypergraph'' is a hypergraph with some edges removed. Given a subset <math>J \subset I_e</math> of the edge index set, the partial hypergraph generated by <math>J</math> is the hypergraph
:<math>\left(X, \lbrace e_i | i\in J \rbrace \right).</math>
Given a subset <math>A\subseteq X</math>, the ''section hypergraph'' is the partial hypergraph
:<math>H \times A = \left(A, \lbrace e_i |
i\in I_e \land e_i \subseteq A \rbrace \right).</math>
The '''dual''' <math>H^*</math> of <math>H</math> is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by <math>\lbrace e_i \rbrace</math> and whose edges are given by <math>\lbrace X_m \rbrace</math> where
:<math>X_m = \lbrace e_i | x_m \in e_i \rbrace. </math>
When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an [[involution (mathematics)|involution]], i.e.,
:<math>\left(H^*\right)^* = H.</math>
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'' [[induced subgraph|induces]] a connected subgraph in ''G''. For a disconnected hypergraph ''H'', ''G'' is a host graph if there is a bijection between the [[connected component (graph theory)|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 ''bipartite'' if and only if its vertices can be partitioned into two classes ''U'' and ''V'' in such a way that each hyperedge with cardinality at least 2 contains at least one vertex from both classes. Alternatively, such a hypergraph is said to have [[Property B]].
The '''2-section''' (or '''clique graph''', '''representing graph''', '''primal graph''', '''Gaifman 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.
==Bipartite graph model==
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. This bipartite graph is also called [[incidence graph]].
==Acyclicity==
In contrast with ordinary undirected graphs for which there is a single natural notion of [[cycle (graph theory)|cycles]] and [[Forest (graph theory)|acyclic graphs]], there are multiple natural non-equivalent definitions of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the special case of ordinary graphs.
A first definition of acyclicity for hypergraphs was given by [[Claude Berge]]:<ref>[[Claude Berge]], ''Graphs and Hypergraphs''</ref> a hypergraph is Berge-acyclic if its [[incidence graph]] (the [[bipartite graph]] defined above) is acyclic. This definition is very restrictive: for instance, if a hypergraph has some pair <math>v \neq v'</math> of vertices and some pair <math>f \neq f'</math> of hyperedges such that <math>v, v' \in f</math> and <math>v, v' \in f'</math>, then it is Berge-cyclic. Berge-cyclicity can obviously be tested in [[linear time]] by an exploration of the incidence graph.
We can define a weaker notion of hypergraph acyclicity,<ref>C. Beeri, [[Ronald Fagin|R. Fagin]], D. Maier, [[Mihalis Yannakakis|M. Yannakakis]], ''On the Desirability of Acyclic Database Schemes''</ref> later termed α-acyclicity. This notion of acyclicity is equivalent to the hypergraph being conformal (every clique of the primal graph is covered by some hyperedge) and its primal graph being [[chordal graph|chordal]]; it is also equivalent to reducibility to the empty graph through the GYO algorithm<ref>C. T. Yu and M. Z. Özsoyoğlu. ''[https://www.computer.org/csdl/proceedings/cmpsac/1979/9999/00/00762509.pdf An algorithm for tree-query membership of a distributed query]''. In Proc. IEEE COMPSAC, pages 306-312, 1979</ref><ref name="graham1979universal">M. H. Graham. ''On the universal relation''. Technical Report, University of Toronto, Toronto, Ontario, Canada, 1979</ref> (also known as Graham's algorithm), a [[confluence (abstract rewriting)|confluent]] iterative process which removes hyperedges using a generalized definition of [[ear (graph theory)|ears]]. In the domain of [[database theory]], it is known that a [[database schema]] enjoys certain desirable properties if its underlying hypergraph is α-acyclic.<ref>[[Serge Abiteboul|S. Abiteboul]], [[Richard B. Hull|R. B. Hull]], [[Victor Vianu|V. Vianu]], ''Foundations of Databases''</ref> Besides, α-acyclicity is also related to the expressiveness of the [[guarded fragment]] of [[first-order logic]].
We can test in [[linear time]] if a hypergraph is α-acyclic.<ref>[[Robert Tarjan|R. E. Tarjan]], [[Mihalis Yannakakis|M. Yannakakis]]. ''Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs''. SIAM J. on Computing, 13(3):566-579, 1984.</ref>
Note that α-acyclicity has the counter-intuitive property that adding hyperedges to an α-cyclic hypergraph may make it α-acyclic (for instance, adding a hyperedge containing all vertices of the hypergraph will always make it α-acyclic). Motivated in part by this perceived shortcoming, [[Ronald Fagin]]<ref name="fagin1983degrees">[[Ronald Fagin]], ''Degrees of Acyclicity for Hypergraphs and Relational Database Schemes''</ref> defined the stronger notions of β-acyclicity and γ-acyclicity. We can state β-acyclicity as the requirement that all subhypergraphs of the hypergraph are α-acyclic, which is equivalent<ref name="fagin1983degrees"/> to an earlier definition by Graham.<ref name="graham1979universal"/> The notion of γ-acyclicity is a more restrictive condition which is equivalent to several desirable properties of database schemas and is related to [[Bachman diagram]]s. Both β-acyclicity and γ-acyclicity can be tested in [[PTIME|polynomial time]].
Those four notions of acyclicity are comparable: Berge-acyclicity implies γ-acyclicity which implies β-acyclicity which implies α-acyclicity. However, none of the reverse implications hold, so those four notions are different.<ref name="fagin1983degrees" />
==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 <math>H=(X,E)</math> is ''isomorphic'' to a hypergraph <math>G=(Y,F)</math>, written as <math>H \simeq G</math> if there exists a [[bijection]]
:<math>\phi:X \to Y</math>
and a [[permutation]] <math>\pi</math> of <math>I</math> such that
:<math>\phi(e_i) = f_{\pi(i)}</math>
The bijection <math>\phi</math> is then called the [[isomorphism]] of the graphs. Note that
:<math>H \simeq G</math> if and only if <math>H^* \simeq G^*</math>.
When the edges of a hypergraph are explicitly labeled, one has the additional notion of ''strong isomorphism''. One says that <math>H</math> is ''strongly isomorphic'' to <math>G</math> if the permutation is the identity. One then writes <math>H \cong G</math>. 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 <math>H</math> is ''equivalent'' to <math>G</math>, and writes <math>H\equiv G</math> if the isomorphism <math>\phi</math> has
:<math>\phi(x_n) = y_n</math>
and
:<math>\phi(e_i) = f_{\pi(i)}</math>
Note that
:<math>H\equiv G</math> if and only if <math>H^* \cong G^*</math>
If, in addition, the permutation <math>\pi</math> is the identity, one says that <math>H</math> equals <math>G</math>, and writes <math>H=G</math>. Note that, with this definition of equality, graphs are self-dual:
:<math>\left(H^*\right) ^* = H</math>
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 (mathematics)|group]] under composition, called the [[automorphism group]] of the hypergraph and written Aut(''H'').
===Examples===
Consider the hypergraph <math>H</math> with edges
:<math>H = \lbrace
e_1 = \lbrace a,b \rbrace,
e_2 = \lbrace b,c \rbrace,
e_3 = \lbrace c,d \rbrace,
e_4 = \lbrace d,a \rbrace,
e_5 = \lbrace b,d \rbrace,
e_6 = \lbrace a,c \rbrace
\rbrace</math>
and
:<math>G = \lbrace
f_1 = \lbrace \alpha,\beta \rbrace,
f_2 = \lbrace \beta,\gamma \rbrace,
f_3 = \lbrace \gamma,\delta \rbrace,
f_4 = \lbrace \delta,\alpha \rbrace,
f_5 = \lbrace \alpha,\gamma \rbrace,
f_6 = \lbrace \beta,\delta \rbrace
\rbrace</math>
Then clearly <math>H</math> and <math>G</math> are isomorphic (with <math>\phi(a)=\alpha</math>, ''etc.''), but they are not strongly isomorphic. So, for example, in <math>H</math>, vertex <math>a</math> meets edges 1, 4 and 6, so that,
:<math>e_1 \cap e_4 \cap e_6 = \lbrace a\rbrace</math>
In graph <math>G</math>, there does not exist any vertex that meets edges 1, 4 and 6:
:<math>f_1 \cap f_4 \cap f_6 = \varnothing</math>
In this example, <math>H</math> and <math>G</math> are equivalent, <math>H\equiv G</math>, and the duals are strongly isomorphic: <math>H^*\cong G^*</math>.
==Symmetric hypergraphs==
The ''{{visible anchor|rank}}'' <math>r(H)</math> of a hypergraph <math>H</math> 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 <math>\phi(x)=y</math>. Two edges <math>e_i</math> and <math>e_j</math> are said to be ''symmetric'' if there exists an automorphism such that <math>\phi(e_i)=e_j</math>.
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.
==Transversals==
A ''[[Transversal (combinatorics)|transversal]]'' (or "[[hitting set]]") of a hypergraph ''H'' = (''X'', ''E'') is a set <math>T\subseteq X</math> that has nonempty [[intersection (set theory)|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 [[combinatorial optimization]], in [[game theory]], and in several fields of [[computer science]] such as [[machine learning]], [[Index (database)|indexing of database]]s, the [[Boolean satisfiability problem|satisfiability problem]], [[data mining]], and computer [[program optimization]].
==Incidence matrix==
Let <math>V = \{v_1, v_2, ~\ldots, ~ v_n\}</math> and <math>E = \{e_1, e_2, ~ \ldots ~ e_m\}</math>. Every hypergraph has an <math>n \times m</math> [[incidence matrix]] <math>A = (a_{ij})</math> where
:<math>a_{ij} = \left\{ \begin{matrix} 1 & \mathrm{if} ~ v_i \in e_j \\ 0 & \mathrm{otherwise}. \end{matrix} \right.</math>
The [[transpose]] <math>A^t</math> of the [[incidence (geometry)|incidence]] matrix defines a hypergraph <math>H^* = (V^*,\ E^*)</math> called the '''dual''' of <math>H</math>, where <math>V^*</math> is an ''m''-element set and <math>E^*</math> is an ''n''-element set of subsets of <math>V^*</math>. For <math>v^*_j \in V^*</math> and <math>e^*_i \in E^*, ~ v^*_j \in e^*_i</math> [[if and only if]] <math>a_{ij} = 1</math>.
==Hypergraph coloring==
Classic hypergraph coloring is assigning one of the colors from set <math>\{1,2,3,...\lambda\}</math> to every vertex of a hypergraph in such a way that each hyperedge contains at least two vertices of distinct colors. In other words, there must be no monochromatic hyperedge with cardinality at least 2. In this sense it is a direct generalization of graph coloring. Minimum number of used distinct colors over all colorings is called the chromatic number of a hypergraph.
Hypergraphs for which there exists a coloring using up to ''k'' colors are referred to as ''k-colorable''. The 2-colorable hypergraphs are exactly the bipartite ones.
There are many generalizations of classic hypergraph coloring. One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. Some mixed hypergraphs are uncolorable for any number of colors. A general criterion for uncolorability is unknown. When a mixed hypergraph is colorable, then the minimum and maximum number of used colors are called the lower and upper chromatic numbers respectively. See http://spectrum.troy.edu/voloshin/mh.html for details.
==Partitions==
A partition theorem due to E. Dauber<ref>E. Dauber, in ''Graph theory'', ed. F. Harary, Addison Wesley, (1969) p. 172.</ref> states that, for an edge-transitive hypergraph <math>H=(X,E)</math>, there exists a [[partition of a set|partition]]
:<math>(X_1, X_2,\cdots,X_K)</math>
of the vertex set <math>X</math> such that the subhypergraph <math>H_{X_k}</math> generated by <math>X_k</math> is transitive for each <math>1\le k \le K</math>, and such that
:<math>\sum_{k=1}^K r\left(H_{X_k} \right) = r(H)</math>
where <math>r(H)</math> 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<ref>{{Citation |title=Multilevel hypergraph partitioning: applications in VLSI domain |author=Karypis, G., Aggarwal, R., Kumar, V., and Shekhar, S. |journal=IEEE Transactions on Very Large Scale Integration (VLSI) Systems |date=March 1999 |volume=7 |issue=1 |pages=69–79 |doi=10.1109/92.748202 |postscript=.|citeseerx=10.1.1.553.2367 }}</ref> and parallel computing.<ref>{{Citation |doi=10.1016/S0167-8191(00)00048-X |title=Graph partitioning models for parallel computing |author= Hendrickson, B., Kolda, T.G. |journal=Parallel Computing | year=2000 |volume=26 |issue=12 |pages=1519–1545 |postscript=.|url=https://digital.library.unt.edu/ark:/67531/metadc684945/ |type=Submitted manuscript }}</ref><ref>{{Cite conference |last1=Catalyurek |first1=U.V. |last2=Aykanat |first2=C. |title=A Hypergraph Model for Mapping Repeated Sparse Matrix-Vector Product Computations onto Multicomputers |conference=Proc. International Conference on Hi Performance Computing (HiPC'95) |year=1995}}</ref><ref>{{Citation |last1=Catalyurek |first1=U.V. |last2=Aykanat |first2=C. |title=Hypergraph-Partitioning Based Decomposition for Parallel Sparse-Matrix Vector Multiplication |journal=IEEE Transactions on Parallel and Distributed Systems |volume=10 |issue=7 |pages=673–693 |year=1999|doi=10.1109/71.780863 |postscript=. |citeseerx=10.1.1.67.2498 }}</ref> Efficient and Scalable [[Graph partition|hypergraph partitioning algorithms]] are also important for processing large scale hypergraphs in machine learning tasks.<ref name=hyperx>{{citation|last1=Huang|first1=Jin|last2=Zhang|first2=Rui|last3=Yu|first3=Jeffrey Xu|journal=Proceedings of the IEEE International Conference on Data Mining|title=Scalable Hypergraph Learning and Processing|year=2015}}</ref>
==Theorems==
Many [[theorem]]s 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.
==Hypergraph drawing==
[[File:CircuitoDosMallas.png|thumb|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.]]
Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs.
In one possible visual representation for hypergraphs, similar to the standard [[graph drawing]] style in which curves in the plane are used to depict graph edges, a hypergraph's vertices are depicted as points, disks, or boxes, and its hyperedges are depicted as trees that have the vertices as their leaves.<ref>{{citation
| last = Sander | first = G.
| contribution = Layout of directed hypergraphs with orthogonal hyperedges
| pages = 381–386
| publisher = Springer-Verlag
| series = [[Lecture Notes in Computer Science]]
| title = Proc. 11th International Symposium on Graph Drawing (GD 2003)
| contribution-url = http://gdea.informatik.uni-koeln.de/585/1/hypergraph.ps
| volume = 2912
| year = 2003| title-link = International Symposium on Graph Drawing
}}.</ref><ref>{{citation
| last1 = Eschbach | first1 = Thomas
| last2 = Günther | first2 = Wolfgang
| last3 = Becker | first3 = Bernd
| issue = 2
| journal = [[Journal of Graph Algorithms and Applications]]
| pages = 141–157
| title = Orthogonal hypergraph drawing for improved visibility
| url = http://jgaa.info/accepted/2006/EschbachGuentherBecker2006.10.2.pdf
| volume = 10
| year = 2006 | doi=10.7155/jgaa.00122}}.</ref> If the vertices are represented as points, the hyperedges may also be shown as smooth curves that connect sets of points, or as [[simple closed curve]]s that enclose sets of points.<ref>{{citation
| last = Mäkinen | first = Erkki
| doi = 10.1080/00207169008803875
| issue = 3
| journal = International Journal of Computer Mathematics
| pages = 177–185
| title = How to draw a hypergraph
| volume = 34
| year = 1990}}.</ref><ref>{{citation
| last1 = Bertault | first1 = François
| last2 = Eades | first2 = Peter | author2-link = Peter Eades
| contribution = Drawing hypergraphs in the subset standard
| doi = 10.1007/3-540-44541-2_15
| pages = 45–76
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| title = Proc. 8th International Symposium on Graph Drawing (GD 2000)
| volume = 1984
| year = 2001| title-link = International Symposium on Graph Drawing
| isbn = 978-3-540-41554-1
| doi-access = free
}}.</ref><ref>{{citation
| last1 = Naheed Anjum | first1 = Arafat
| last2 = Bressan | first2 = Stéphane
| contribution = Hypergraph Drawing by Force-Directed Placement
| doi = 10.1007/978-3-319-64471-4_31
| pages = 387–394
| publisher = Springer International Publishing
| series = Lecture Notes in Computer Science
| title = 28th International Conference on Database and Expert Systems Applications (DEXA 2017)
| volume = 10439
| year = 2017| isbn = 978-3-319-64470-7
}}.</ref>
[[File:Venn's four ellipse construction.svg|thumb|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).]]
In another style of hypergraph visualization, the subdivision model of hypergraph drawing,<ref>{{citation
| last1 = Kaufmann | first1 = Michael
| last2 = van Kreveld | first2 = Marc
| last3 = Speckmann | first3 = Bettina | author3-link = Bettina Speckmann
| contribution = Subdivision drawings of hypergraphs
| doi = 10.1007/978-3-642-00219-9_39
| pages = 396–407
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| title = Proc. 16th International Symposium on Graph Drawing (GD 2008)
| volume = 5417
| year = 2009| title-link = International Symposium on Graph Drawing
| isbn = 978-3-642-00218-2
| doi-access = free
}}.</ref> the plane is subdivided into regions, each of which represents a single vertex of the hypergraph. 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. 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). In contrast with the polynomial-time recognition of [[planar graph]]s, it is [[NP-complete]] to determine whether a hypergraph has a planar subdivision drawing,<ref>{{citation
| last1 = Johnson | first1 = David S. | author1-link = David S. Johnson
| last2 = Pollak | first2 = H. O.
| doi = 10.1002/jgt.3190110306
| issue = 3
| journal = Journal of Graph Theory
| pages = 309–325
| title = Hypergraph planarity and the complexity of drawing Venn diagrams
| volume = 11
| year = 2006}}.</ref> 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.<ref>{{citation
| last1 = Buchin | first1 = Kevin
| last2 = van Kreveld | first2 = Marc
| last3 = Meijer | first3 = Henk
| last4 = Speckmann | first4 = Bettina
| last5 = Verbeek | first5 = Kevin
| contribution = On planar supports for hypergraphs
| doi = 10.1007/978-3-642-11805-0_33
| pages = 345–356
| publisher = Springer-Verlag
| series = Lecture Notes in Computer Science
| title = Proc. 17th International Symposium on Graph Drawing (GD 2009)
| volume = 5849
| year = 2010| title-link = International Symposium on Graph Drawing
| isbn = 978-3-642-11804-3
| doi-access = free
}}.</ref>
An alternative representation of the hypergraph called PAOH<ref name="paoh" /> is shown in the figure on top of this article. Edges are vertical lines connecting vertices. Vertices are aligned on the left. The legend on the right shows the names of the edges. It has been designed for dynamic hypergraphs but can be used for simple hypergraphs as well.
==Hypergraph grammars==
{{main|Hypergraph grammar}}
By augmenting a class of hypergraphs with replacement rules, [[graph grammar]]s can be generalised to allow hyperedges.
==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, subsets of subsets of vertices and so on ''ad infinitum''. In essence, every edge is just an internal node of a tree or [[directed acyclic graph]], and vertices are the leaf nodes. A hypergraph is then just a collection of trees with common, shared nodes (that is, a given internal node or leaf may occur in several different trees). Conversely, every collection of trees can be understood as this generalized hypergraph. Since trees are widely used throughout [[computer science]] and many other branches of mathematics, one could say that hypergraphs appear naturally as well. So, for example, this generalization arises naturally as a model of [[term algebra]]; edges correspond to [[term (logic)|terms]] and vertices correspond to constants or variables.
For such a hypergraph, 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 <math>V= \{a,b\}</math> and whose edges are <math>e_1=\{a,b\}</math> and <math>e_2=\{a,e_1\}</math>. Then, although <math>b\in e_1</math> and <math>e_1\in e_2</math>, it is not true that <math>b\in e_2</math>. 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 <math>e_1</math> and <math>e_2</math>, and zero vertices, so that <math>e_1 = \{e_2\}</math> and <math>e_2 = \{e_1\}</math>. 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 graph|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
:<math>\left[ \begin{matrix} 0 & 1 \\ 1 & 0 \end{matrix} \right].</math>
==Hypergraph learning==
Hypergraphs have been extensively used in [[machine learning]] tasks as the data model and classifier [[regularization (mathematics)]].<ref>{{citation| last1 = Zhou | first1 = Dengyong| last2 = Huang | first2 = Jiayuan | last3=Scholkopf | first3=Bernhard| issue = 2| journal = Advances in Neural Information Processing Systems| pages = 1601–1608| title = Learning with hypergraphs: clustering, classification, and embedding| year = 2006}}</ref> The applications include [[recommender system]] (communities as hyperedges),<ref>{{citation|last1=Tan | first1=Shulong | last2=Bu | first2=Jiajun | last3=Chen | first3=Chun | last4=Xu | first4=Bin | last5=Wang | first5=Can | last6=He | first6=Xiaofei|issue = 1| journal = ACM Transactions on Multimedia Computing, Communications, and Applications| title = Using rich social media information for music recommendation via hypergraph model| year = 2013|url=https://www.researchgate.net/publication/226075153| bibcode=2011smma.book..213T }}</ref> [[image retrieval]] (correlations as hyperedges),<ref>{{citation|last1=Liu | first1=Qingshan | last2=Huang | first2=Yuchi | last3=Metaxas | first3=Dimitris N. |issue = 10–11| journal = Pattern Recognition| title = Hypergraph with sampling for image retrieval| pages=2255–2262| year = 2013| doi=10.1016/j.patcog.2010.07.014 | volume=44}}</ref> and [[bioinformatics]] (biochemical interactions as hyperedges).<ref>{{citation|last1=Patro |first1=Rob | last2=Kingsoford | first2=Carl| issue = 10–11| journal = Bioinformatics| title = Predicting protein interactions via parsimonious network history inference| year = 2013| pages=237–246|doi=10.1093/bioinformatics/btt224 |pmid=23812989 |pmc=3694678 | volume=29}}</ref> Representative hypergraph learning techniques include hypergraph [[spectral clustering]] that extends the [[spectral graph theory]] with hypergraph Laplacian,<ref>{{citation|last1=Gao | first1=Tue | last2=Wang | first2=Meng | last3=Zha|first3=Zheng-Jun|last4=Shen|first4=Jialie|last5=Li|first5=Xuelong|last6=Wu|first6=Xindong|issue = 1| journal = IEEE Transactions on Image Processing| volume=22 | title = Visual-textual joint relevance learning for tag-based social image search| year = 2013| pages=363–376|url=http://ink.library.smu.edu.sg/cgi/viewcontent.cgi?article=2510&context=sis_research | doi=10.1109/tip.2012.2202676| pmid=22692911 | bibcode=2013ITIP...22..363Y }}</ref> and hypergraph [[semi-supervised learning]] that introduces extra hypergraph structural cost to restrict the learning results.<ref>{{citation|last1=Tian|first1=Ze|last2=Hwang|first2=TaeHyun|last3=Kuang|first3=Rui|issue = 21| journal = Bioinformatics| title = A hypergraph-based learning algorithm for classifying gene expression and arrayCGH data with prior knowledge| year = 2009| pages=2831–2838|doi=10.1093/bioinformatics/btp467|pmid=19648139| volume=25|doi-access=free}}</ref> For large scale hypergraphs, a distributed framework<ref name=hyperx /> built using [[Apache Spark]] is also available.
==See also==
{{Commons category|Hypergraphs}}
* [[Simplicial complex]]
* [[Combinatorial design]]
* [[Factor graph]]
* [[Greedoid]]
* [[Incidence structure]]
* [[Matroid]]
* [[Multigraph]]
* [[P system]]
* [[Sparse matrix-vector multiplication]]
*[[Matching in hypergraphs]]
==Notes==
{{Reflist}}
==References==
* Claude Berge, "Hypergraphs: Combinatorics of finite sets". North-Holland, 1989.
* Claude Berge, Dijen Ray-Chaudhuri, "Hypergraph Seminar, Ohio State University 1972", ''Lecture Notes in Mathematics'' '''411''' Springer-Verlag
* {{springer|title=Hypergraph|id=p/h048470}}
* Alain Bretto, "Hypergraph Theory: an Introduction", Springer, 2013.
* Vitaly I. Voloshin. "Coloring Mixed Hypergraphs: Theory, Algorithms and Applications". Fields Institute Monographs, American Mathematical Society, 2002.
* Vitaly I. Voloshin. "Introduction to Graph and Hypergraph Theory". [[Nova Science Publishers, Inc.]], 2009.
* {{PlanetMath attribution|urlname=hypergraph|title=hypergraph}}
==External links==
* [https://www.aviz.fr/paohvis PAOHVis]: open-source PAOHVis system for visualizing dynamic hypergraphs.
{{Graph representations}}
[[Category:Hypergraphs| ]]
[[de:Graph (Graphentheorie)#Hypergraph]]