Donor challenge: Your generous donation will be matched 2-to-1 right now. Your $5 becomes$15! Dear Internet Archive Supporter,. I ask only. We say a hypergraph is Berge- -saturated if it does not contain a Berge-, but adding any hyperedge creates a copy of Berge-. The -uniform. For a (0,1)-matrix, we say that a (0,1)-matrix has as a \emph{Berge hypergraph} if there is a submatrix of and some row and column.

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## Hypergraph

However, none of the reverse implications hold, so those four notions are different. Thus, for the above example, the incidence matrix is simply. By using this site, you agree to the Terms of Use and Privacy Policy.

A partition theorem due to E. A hypergraph automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices.

### Hypergraph – Wikipedia

This definition is very restrictive: A first definition of acyclicity for hypergraphs was given by Claude Berge: For such a hypergraph, set membership then provides an ordering, but the ordering is neither a partial order nor a preordersince it is not transitive. 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.

Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above. On the universal relation. The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms. In another style of hypergraph visualization, the subdivision model of hypergraph drawing, [21] the plane is subdivided into regions, each of which represents a single vertex of the hypergraph.

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When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism. Note that, with this definition of equality, graphs are self-dual:.

The graph corresponding to the Levi graph of this generalization is a directed acyclic graph. Minimum number of used distinct colors over all colorings is called the chromatic number of a hypergraph. As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation. There are two variations of this generalization.

## Mathematics > Combinatorics

This page was last edited on 27 Decemberat Computing the transversal hypergraph has applications in combinatorial optimizationin game theoryand in several fields of computer science such as machine learningindexing of databasesthe satisfiability problemhypetgraphs miningand computer program optimization. If all edges have the same cardinality kthe hypergraph is said to be uniform or k -uniformor is called a k -hypergraph.

Similarly, a hypergraph is edge-transitive if all edges are symmetric. Harary, Addison Wesley, p. Those four notions of acyclicity are comparable: In other words, there must be no monochromatic hyperedge with cardinality at least 2.

Some mixed hypergraphs are uncolorable for any number of colors. In computational geometrya hypergraph may sometimes be called a range space hypfrgraphs then the hyperedges are called ranges.

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. A general criterion for uncolorability is unknown. Special kinds of hypergraphs include: While graph edges are pairs of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes.

Some methods for studying symmetries of graphs extend to hypergraphs. Hypergraphs have been extensively used in machine learning tasks as the data model and classifier regularization mathematics.

In other projects Wikimedia Commons. There are variant definitions; sometimes edges must not be empty, and sometimes multiple edges, with the same set of nodes, are allowed. Alternatively, such a hypergraph is said to have Property B.

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A hypergraph is said to be vertex-transitive or vertex-symmetric if all of its vertices are symmetric. Hypergraphs have many other names. So, for example, this generalization arises naturally as a model of term algebra ; edges correspond to terms and vertices correspond to constants or variables. When the vertices of a hypergraph are explicitly labeled, one has the notions of equivalenceand also of equality.

In particular, there is a bipartite “incidence graph” or ” Levi graph ” corresponding to every hypergraph, and conversely, most, but not all, bipartite graphs can be regarded as incidence graphs of hypergraphs.

### [] Linearity of Saturation for Berge Hypergraphs

Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs. However, the transitive closure of set membership for such hypergraphs does induce a partial orderand “flattens” the hypergraph into a partially ordered set.

A hypergraph H may be represented by a bipartite graph BG as follows: So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on.

In contrast with ordinary undirected graphs for which there is a single natural notion of cycles and acyclic graphsthere are multiple natural non-equivalent ebrge of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the hypergrapns case of ordinary graphs.

In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes. Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph. 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 hyperraphs a connected subgraph in G.

The 2-colorable hypergraphs are exactly the bipartite ones.