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 definition is very restrictive: Generalization of graph theory. A partition theorem due to E.

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. When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involutioni. Yypergraphs possible generalization of a hypergraph is to allow edges to point at other edges.

### Hypergraph – Wikipedia

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 Sperner theory.

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. Note that all strongly isomorphic graphs are isomorphic, but not vice versa. A hypergraph homomorphism is nypergraphs map from the vertex set of one hypergraph to another such that each edge maps to one other edge.

However, it is often desirable to study hypergraphs where all hyperedges have the same cardinality; a hypergrapsh – uniform hypergraph is a hypergraph such that all its hyperedges have size k. A hypergraph automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices.

## Mathematics > Combinatorics

Alternatively, such a hypergraph is said to have Property B. Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph. 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.

Many theorems and concepts involving graphs also hold for hypergraphs.

### [] Linearity of Saturation for Berge Hypergraphs

Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above. The generalized incidence matrix for such hypergraphs is, by definition, a square matrix, of a rank equal to the total number of vertices plus edges.

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 hypedgraphs as Graham’s algorithma confluent iterative process which removes hyperedges using a generalized definition of ears. The hyperedges of the hypergraph are hyppergraphs by contiguous subsets of these regions, which may be indicated by coloring, by drawing outlines around them, or both.

When a mixed hypergraph is colorable, then the minimum and maximum number of used colors are called the lower and upper chromatic numbers respectively. Ramsey’s theorem and Line graph of a hypergraph are typical examples.

H is k -regular if every vertex has degree k. 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.

Some methods for studying symmetries of graphs extend to hypergraphs. One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. In one, the edges consist not only of a set of vertices, but may also contain subsets bergf vertices, subsets of subsets of vertices and so on ad infinitum. Since trees are widely used throughout computer science and many other branches of mathematics, one could say that hypergraphs appear naturally as well.

Dauber, in Graph theoryed. A hypergraph is then just a collection of trees with common, shared nodes that is, a given internal node or leaf may occur in several different trees.

On the universal relation. The 2-section or clique graphrepresenting graphprimal graphBwrge 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. Retrieved from ” https: 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. In this sense it is a direct generalization of graph coloring.

So, for example, this generalization arises naturally as a model of term algebra ; edges correspond to terms and vertices correspond to constants or variables. Harary, Addison Wesley, p.

This page was last edited on 27 Decemberat The degree d v of a vertex v is the number of edges that contain it. Special kinds of hypergraphs include: Simple linear-time algorithms to test chordality of graphs, test acyclicity of hypergraphs, and selectively reduce acyclic hypergraphs. There are two variations of this generalization. If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply transitive.

Because hypergraph links can have any cardinality, there are several notions of the concept of a subgraph, called subhypergraphspartial hypergraphs and section hypergraphs. Note that, with this definition of equality, graphs are self-dual:. So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. Similarly, a hypergraph is edge-transitive if all edges are symmetric. 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 is also called a set system or a family of sets drawn from the universal set X.

From Wikipedia, the free encyclopedia. Hypergraphs have been extensively used in machine learning tasks as the data model and classifier regularization mathematics. 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’. Those four notions of acyclicity are comparable: As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation.