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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Wikimedia Commons has media related to Hypergraphs. Dauber, in Graph theoryed. The 2-section or clique graphrepresenting graphprimal graphGaifman graph of a hypergraph is the graph with the same vertices of the hypergraph, and hypergarphs between all pairs of vertices contained in the same hyperedge.

When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involutioni. H is k -regular if every vertex has degree k. 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 other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes.

## Graphs And Hypergraphs

In another style of hypergraph visualization, the subdivision model of hypergraph nerge, [21] the plane is subdivided into regions, each of which represents a single vertex of the hypergraph. A hypergraph automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices. Since trees are widely used throughout computer science and many other branches of mathematics, one could say that hypergraphs appear naturally as well.

When the vertices of a hypergraph are explicitly labeled, one has the notions of equivalenceand also of equality.

### Hypergraph – Wikipedia

Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. By augmenting a class of hypergraphs with replacement rules, graph grammars can be generalised to allow hyperedges.

In computational geometrya hypergraph may sometimes be called a range space and then the hyperedges are called ranges. This notion of acyclicity is equivalent to the hypergraph being conformal every clique of the primal graph is covered by some hyperedge and its primal graph being chordal ; it is also equivalent to reducibility to the empty graph through the GYO algorithm [5] [6] also known as Graham’s algorithma confluent iterative process which removes hyperedges using a generalized definition of ears.

As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation. The difference between a set system and a hypergraph is in the questions being asked.

In one, the edges consist not only of a set of vertices, but may also contain subsets of vertices, subsets of subsets of vertices and so on ad infinitum. In other projects Wikimedia Commons. Those four notions of acyclicity are comparable: One possible generalization of a hypergraph is to allow edges to point at other edges.

One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. Computing the transversal hypergraph has applications in combinatorial optimizationin game bbergeand in several fields of computer science such as machine learningindexing of hypergraphhsthe satisfiability problemdata miningand computer program optimization.

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### [] Forbidden Berge Hypergraphs

Harary, Addison Wesley, p. Hypergraphs can be viewed as incidence structures. In contrast with ordinary undirected graphs for which there is a single natural notion of cycles and acyclic graphsthere are multiple natural non-equivalent definitions of acyclicity for hypergraphs which collapse to ordinary graph acyclicity for the special herge of ordinary graphs. Special kinds of hypergraphs include: A general criterion for uncolorability is unknown. 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.

Similarly, a hypergraph is edge-transitive hypergaphs all edges are symmetric. A hypergraph is bipartite if and only if its vertices can be partitioned into two classes U and V in such a way that each hyperedge with cardinality at least 2 contains at least one vertex from both classes. Herge degree d v of hypergralhs vertex v is the number of edges that contain it.

If all edges have the same cardinality kthe hypergraph is said to be uniform or k -uniformor is called a k -hypergraph.

## Mathematics > Combinatorics

When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism. Note that all strongly isomorphic graphs are isomorphic, but not vice versa. There are variant definitions; sometimes edges must not be empty, and sometimes multiple edges, with the same set hypergraohs nodes, are allowed.

Hypergraph theory tends to concern questions similar to those of graph theory, such as connectivity and colorabilitywhile the theory of set systems tends to ask non-graph-theoretical questions, such as those of Bergs theory. There are two variations of this generalization.

From Wikipedia, the free encyclopedia. For a disconnected hypergraph HG is a host graph if there is a bijection between the connected components of G and of Hsuch that each connected component G’ of G is a host of the corresponding H’. The bergs 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.

So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. In mathematicsa hypergraph is a generalization of a graph in which an edge can join any number of vertices. In some literature edges are referred to as hyperlinks or connectors. Retrieved from ” https: Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs.