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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If a hypergraph is both edge- and vertex-symmetric, then the hypergraph is simply transitive. In some literature edges are referred to as hyperlinks or connectors. 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. A first definition of acyclicity for hypergraphs was given by Claude Berge: Many theorems and concepts involving graphs also hold for hypergraphs. 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.

Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above. 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.


One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. In essence, every edge is just an internal node of a tree or directed acyclic graphand vertices are the leaf nodes. A partition theorem due to E. On the universal relation.

However, none of the reverse implications hold, so those four notions are different. Hypergraphs have been extensively used in machine learning tasks as the data model and classifier regularization mathematics.


Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity. Those four notions of acyclicity are comparable: This bipartite graph is also called incidence graph.

A hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge. However, the transitive closure of set membership for such hypergraphs does induce a partial orderand “flattens” the hypergraph into a partially ordered set. By using this site, you agree to the Terms of Use and Privacy Policy. When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism.

So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph.

Graph partitioning and in particular, hypergraph partitioning has many applications to IC design [11] and parallel computing. The 2-colorable hypergraphs are exactly bfrge bipartite ones.

Graphs And Hypergraphs

The degree d v of a vertex v is the number of edges that contain it. 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 case of ordinary graphs.

Dauber, in Graph theoryed. This page was last edited on 27 Decemberat 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 hypefgraphs hyperedges are depicted as trees that have the vertices as their leaves.

In particular, there is no transitive closure of set membership for such hypergraphs. 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.

Harary, Addison Wesley, p. A hypergraph automorphism is an isomorphism from a vertex set into itself, that is a relabeling of vertices.


Note that, with this definition of equality, graphs hypegrraphs self-dual:. A transversal T is called minimal if no proper subset of T is a transversal.

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.

Hypergraphs for which there hyppergraphs a coloring using up to k colors are referred to as k-colorable. Some mixed hypergraphs are uncolorable for any number of colors. There are variant definitions; sometimes edges must not be empty, and sometimes multiple edges, with the same set of nodes, are allowed. There are two variations of this generalization. Wikimedia Commons has media related to Hypergraphs.

Hypergraph – Wikipedia

Alternately, edges can be allowed to point at other edges, irrespective of the requirement that the edges be ordered as directed, acyclic graphs. One possible generalization of a hypergraph is to allow edges to point at other edges. An algorithm for tree-query membership of a distributed hypergraps. When a mixed hypergraph is colorable, then the minimum and maximum number of used colors are called the lower hyperyraphs upper chromatic numbers respectively.

The difference between a set system and a hypergraph is in the questions being asked. In computational geometrya hypergraph may sometimes be called a range space and then the hyperedges are called ranges.

March”Multilevel hypergraph partitioning: 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.

In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes. A hypergraph is also called a set system or a family of sets drawn from the universal set X.