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 , and is uniform if it is -uniform for some . An ordinary graph is merely a uniform hypergraph.
The degree of a vertex is the number of edges in that contain this vertex, often denoted . is -regular if every vertex has degree , and is said to be regular if it is -regular for some .
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.
|Date of creation||2013-03-22 13:05:29|
|Last modified on||2013-03-22 13:05:29|
|Last modified by||CWoo (3771)|