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. Alternatively, such a hypergraph is said to have Property B. The 2-section or clique graph, representing graph, primal graph, Gaifman 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. Bipartite graph model[ edit ] A hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BG, and x1, e1 are connected with an edge if and only if vertex x1 is contained in edge e1 in H.
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Mikabei A hypergraph is also called a set system or a family of sets drawn from the universal set X. The degree d v of a vertex v is the number of edges that contain it.
The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms. A first definition of acyclicity for hypergraphs was given by Claude Berge: When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involutioni. One possible generalization of a hypergraph is to allow edges to point at other edges. In another style of berve visualization, the subdivision model of hypergraph drawing,  the plane is subdivided into regions, each of which represents a single vertex of the hypergraph.
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. This page was last edited on 27 Decemberat Minimum number of used distinct colors over all colorings is called the chromatic number of a hypergraph.
When the vertices of a hypergraph are hypergraphhs labeled, one has the notions of equivalenceand also of equality. Special kinds of hypergraphs include: Many theorems and concepts involving graphs also hold for hypergraphs. The 2-colorable hypergraphs are exactly the bipartite ones. Hypergraph — Wikipedia Computing the transversal hypergraph has applications in combinatorial optimizationin game theoryand in several fields of computer science hyperbraphs as machine learningindexing of databasesthe satisfiability problemdata miningand computer program optimization.
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.
Hypergraphs hypergrapha which there exists a coloring using up to k colors are referred to as k-colorable. The partial hypergraph is a hypergraph with some edges removed. There are two variations of this generalization. Note that all strongly isomorphic graphs are bypergraphs, but not vice versa.
There are many generalizations of classic hypergraph coloring. A hypergraph is said to be vertex-transitive or vertex-symmetric if all of its hypergrapns are symmetric. H is k -regular if every vertex has degree k. Graph partitioning and in particular, hypergraph partitioning has many applications to IC design  and parallel computing. The difference between a set system and a hypergraph is in the questions being asked. In other words, there must be no monochromatic hyperedge with cardinality at least 2.
One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. Because of hypergraph duality, the study of edge-transitivity is identical to the study of vertex-transitivity.
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. A subhypergraph is a hypergraph with some vertices removed. When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism.
If a hypergraph is both edge- and vertex-symmetric, then the bergr is simply transitive. 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. Graphs And Hypergraphs Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph. A general criterion for uncolorability is unknown.
Claude was the second of six children; his siblings were Nicole, Antoine, Philippe, Edith and Patrick. In this paper, Berge examines properties of games where there is perfect information available and there are infinite choices for each move. This thesis served as the basis of a page paper published in From to he directed the International Computing Center in Rome. He held visiting positions at Princeton University in , Pennsylvania State University in , and New York University in , and was a frequent visitor to the Indian statistical institute , Calcutta. These books helped bring the subjects of graph theory and combinatorics out of disrepute by highlighting the successful practical applications of the subjects. Art[ edit ] In addition to mathematics, Claude Berge enjoyed literature, sculpture, and art.
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Graphs And Hypergraphs
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