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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This bipartite graph is also called incidence graph. Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called subhypergraphspartial hypergraphs and section hypergraphs. One possible generalization of a hypergraph is to allow edges to point at other edges. In computational geometrya hypergraph may sometimes be called a range space and then the hyperedges are called ranges.

A graph is just a 2-uniform hypergraph. In other projects Wikimedia Commons. 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 bipartitebut is rather just some general directed graph.

Computing the transversal hypergraph has applications in combinatorial optimizationin game theoryand in several fields of computer science such as machine learningindexing of databasesthe satisfiability problemdata miningand computer program optimization. Note that all strongly isomorphic graphs are isomorphic, but not vice versa. A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge.

In particular, there is no transitive closure of set membership for such hypergraphs. Wikimedia Commons has media related to Hypergraphs. 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.

While graph edges are pairs of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. 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.

Hypergraphs have been extensively used in machine learning tasks as the data model and classifier regularization mathematics. Since trees are widely used throughout computer science and many other branches of mathematics, one could say that hypergraphs appear naturally as well.

The transversal hypergraph of H is the hypergraph XF whose edge set F consists of all minimal transversals of H. Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity. 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.

There are two variations of this generalization. Similarly, a hypergraph is edge-transitive if all edges are symmetric. Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs. Some mixed hypergraphs are uncolorable for any number of colors. 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.

The graph corresponding to the Levi graph of this generalization is a directed acyclic graph. There are variant definitions; sometimes edges must not be empty, and sometimes multiple edges, with the same set of nodes, are allowed.

## Hypergraph

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. Simple linear-time beerge to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. An algorithm for tree-query membership of a distributed query.

The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.

### Hypergraph – Wikipedia

In another style of hypergraph visualization, the verge model of hypergraph drawing, [21] the plane is subdivided into regions, each of which represents a single vertex of the hypergraph. Those four notions of acyclicity are comparable: A hypergraph is said to be vertex-transitive or vertex-symmetric if all of its vertices are symmetric.

The 2-section or clique graphrepresenting graphprimal graphGaifman 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. This definition is very restrictive: By augmenting a class of hypergraphs with replacement rules, graph grammars can be generalised to allow hyperedges.

In essence, every edge is just an internal node of a tree or directed acyclic graphand vertices are the leaf nodes. Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph. Views Read Edit View history. Some methods for studying symmetries of graphs extend to hypergraphs.

This allows graphs with edge-loops, which need not contain vertices at all. Special kinds of hypergraphs include: On the universal relation. In contrast with the polynomial-time recognition of planar graphsit is NP-complete to determine whether a hypergraph has a planar subdivision drawing, [22] 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.

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 induces a connected subgraph in G. Dauber, in Graph theoryed. However, the transitive closure of set membership for such hypergraphs does induce a partial orderand “flattens” the hypergraph into a partially ordered set. Note that, with this definition of equality, graphs are bwrge.

There are many generalizations of classic hypergraph coloring. From Wikipedia, the free encyclopedia. A hypergraph is then just a collection of trees with common, shared nodes that is, a given internal node hyperhraphs leaf may occur in several different trees. The partial hypergraph is a hypergraph with some edges removed.