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A hypergraph
is an ordered pair
where is a set of vertices and
is a set of edges such that
. In other words, an edge is nothing more than a set of vertices. A hypergraph is the same thing as a simple incidence structure, but with terminology that focuses on the relationship with graphs.
Sometimes it is desirable to restrict this definition more. The empty hypergraph is not very interesting, so we usually accept that
. Singleton edges are allowed in general, but not the empty one. Most applications consider only finite hypergraphs, but occasionally it is also useful to allow to be infinite.
Many of the definitions of graphs carry verbatim to hypergraphs.
is said to be -uniform if every edge
has cardinality . The degree of a vertex is the number of edges in
that contain this vertex, often denoted . is -regular if every vertex has degree . Notice that an ordinary graph is merely a uniform hypergraph.
Let
and
. Associated to any hypergraph is the
incidence matrix
where
For example, let
, where
and
. Defining and in the obvious manner (as they are listed in the sets), we have
Notice that the sum of the entries in any column is the cardinality of the corresponding edge. Similarly, the sum of the entries in a particular row is the degree of the corresponding vertex.
The transpose of the incidence matrix also defines a hypergraph
, the dual of
, in an obvious manner. To be explicit, let
where is an -element set and
is an -element set of subsets of . For
and
if and only if
.
Continuing from the example above, the dual
of
consists of
and
, where and are defined in the obvious manner.
By the remark immediately after the definition of the incidence matrix of a hypergraph, it is easy to see that the dual of a uniform hypergraph is regular and vice-versa. It is not rare to see fruitful results emerge by considering the dual of a hypergraph.
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"hypergraph" is owned by CWoo. [ full author list (6) | owner history (6) ]
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(view preamble)
Cross-references: subsets, transpose, row, column, sum, obvious, regular, contain, number, vertex, degree, cardinality, definitions, infinite, finite, singleton, graphs, incidence structure, simple, edges, vertices, ordered pair
There are 6 references to this entry.
This is version 14 of hypergraph, born on 2002-10-07, modified 2007-08-17.
Object id is 3508, canonical name is Hypergraph.
Accessed 20008 times total.
Classification:
| AMS MSC: | 05C65 (Combinatorics :: Graph theory :: Hypergraphs) |
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Pending Errata and Addenda
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