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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In particular, there is no transitive closure of set membership for such hypergraphs. There are variant definitions; sometimes edges must not be empty, and sometimes multiple edges, with the same set of nodes, are allowed.

Harary, Addison Wesley, p. Since trees are widely used throughout computer science and many other branches of mathematics, one could say that hypergraphs appear naturally as well. Hypergraphs for which there bedge a coloring using up to k colors are referred to as k-colorable.

[] Forbidden Berge Hypergraphs

Hypergraphs can be viewed as incidence structures. 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.

By augmenting a class of hypergraphs with replacement rules, graph grammars can be generalised to allow hyperedges. Note that all strongly isomorphic graphs are isomorphic, but not vice versa. 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.

In other projects Wikimedia Commons. So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of unordered triples, and so on. Thus, for the above example, the incidence matrix is simply.

As this loop is infinitely recursive, sets that are the edges violate the axiom of foundation. 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. The transversal hypergraph of H is the hypergraph XF whose edge set F consists of all minimal transversals of H. Wikimedia Commons has media related to Hypergraphs.


Many theorems and concepts involving graphs also hold for hypergraphs. Some methods for studying symmetries of graphs extend to hypergraphs.

Computing the transversal hypergraph hypegraphs applications in combinatorial optimizationin game theoryand in several fields of computer science such as machine learningindexing of databasesthe satisfiability problemdata miningand computer program optimization.

The degree d v of a vertex v is the number of edges that contain it. Note that, with this hypergrzphs of equality, graphs are self-dual:. In other words, one such hypergraph is a collection of sets, each such set a hyperedge connecting k nodes. When a mixed hypergraph is colorable, then the minimum and maximum number of used colors are called the lower and upper chromatic numbers respectively.

Mathematics > Combinatorics

The graph corresponding to the Levi graph of this generalization is a directed acyclic graph. 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.

A hypergraph is said to be vertex-transitive or vertex-symmetric if all of its vertices are symmetric. A general criterion for uncolorability is unknown. There are many generalizations of classic hypergraph coloring. This bipartite graph is also called incidence graph.

One of them is the so-called mixed hypergraph coloring, when monochromatic edges are allowed. In some literature edges are referred to as hyperlinks or connectors. 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.


When the vertices of a hypergraph are explicitly labeled, one has the notions of equivalenceand also of equality. When a notion of equality is properly defined, as bergf below, the operation of taking the dual of a hypergraph is an involutioni.


In other words, there must be no monochromatic hyperedge with cardinality at least 2. Hypergraphs have many other names.

The partial hypergraph is a hypergraph with some edges removed. Hypergraphs have been extensively used in machine learning tasks as the data model and classifier regularization mathematics. Although hypergraphs are more difficult to draw on paper than graphs, several researchers have studied methods for the visualization of hypergraphs.

Retrieved from ” https: 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. From Wikipedia, the free encyclopedia.

However, none of the reverse implications hold, so those four notions are different. Similarly, a hypergraph is edge-transitive if all edges are symmetric. A first definition of acyclicity for hypergraphs was given by Claude Berge: 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.

[] Linearity of Saturation for Berge Hypergraphs

A hypergraph H may be represented by a bipartite graph BG as follows: An algorithm for tree-query membership of a distributed query. Hypergraph theory tends to concern questions similar to those of graph theory, bere as connectivity and colorabilitywhile the theory of set systems tends to ask non-graph-theoretical questions, such as hypergtaphs of Sperner theory.

In computational geometrya hypergraph may sometimes be called a range space and then the hyperedges are called ranges. Berge-cyclicity can obviously be tested in linear time by an exploration of the incidence graph.

A subhypergraph is a hypergraph with some vertices removed. Generalization of graph theory.