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Hypergraph

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A generalization of the concept of a graph. A hypergraph is defined by a set , whose elements are known as vertices, and by a family of subsets of , known as edges or hyperedges. A hypergraph is denoted by . The concept of a hypergraph is a variant of the familiar concepts of a complex, a block design and a network.

Two vertices of a hypergraph are said to be adjacent if there exists an edge containing these vertices. A vertex and an edge of a hypergraph are said to be incident if . A hypergraph with vertices and edges may be defined by an incidence matrix, i.e. by the matrix of dimension in which the columns correspond to the edges while the rows correspond to the vertices of the hypergraph, and where

It is possible to assign to each matrix consisting of zeros and ones a hypergraph for which is the incidence matrix. The hypergraph is called the dual of the hypergraph if the incidence matrix of is obtained by transposing the incidence matrix of . The number of edges of a hypergraph that are incident to a given vertex is called the degree of the vertex. The degree of an edge is the number of vertices of the hypergraph incident to this edge. A hypergraph is called a subhypergraph of a hypergraph if , , and if a vertex from and an edge from are incident in the hypergraph if and only if they are incident in the hypergraph .

A hypergraph may be represented in a plane by identifying the vertices of the hypergraph with points of the plane and by identifying the edges with connected domains containing the vertices incident with these edges. For instance, it is possible to represent the hypergraph with set of vertices and family of edges

in a plane, as shown in the figure.

Figure: h048470a

A hypergraph may be represented by a bipartite graph (cf. Graph, bipartite) in which the vertices of one part correspond to the vertices of the hypergraph, while the vertices of the other part correspond to the edges of . Two vertices from and from will be connected by an edge in the graph if the vertex of the hypergraph corresponding to is incident to the edge of the hypergraph corresponding to . A hypergraph is a graph if each of its edges has degree two. An important special case of the concept of a "hypergraph" is that of a matroid. Many concepts in the theory of graphs, such as connectedness, planarity, the chromatic number, and the external and internal stability numbers, may be applied to hypergraphs, as may many theorems that are applicable to graphs.

References

[1] A.A. Zykov, "Hypergraphs" Russian Math. Surveys , 29 : 6 (1974) pp. 89–156 Uspekhi Mat. Nauk , 29 : 6 (1974) pp. 80–154
[2] C. Berge, "Graphs and hypergraphs" , North-Holland (1973) (Translated from French)
How to Cite This Entry:
Hypergraph. A.A. Sapozhenko (originator), Encyclopedia of Mathematics. URL: http://www.encyclopediaofmath.org/index.php?title=Hypergraph&oldid=11526
This text originally appeared in Encyclopedia of Mathematics - ISBN 1402006098