![]() Note that all strongly isomorphic graphs are isomorphic, but not vice-versa. One says that is strongly isomorphic to if the permutation is the identity. When the edges of a hypergraph are explicitly labeled, one has the additional notion of strong isomorphism. The bijection is then called the isomorphism of the graphs. Isomorphism and equalityĪ hypergraph homomorphism is a map from the vertex set of one hypergraph to another such that each edge maps to one other edge.Ī hypergraph is isomorphic to a hypergraph, written as if there exists a bijection This bipartite graph is also called incidence graph. Conversely, any bipartite graph with fixed parts and no unconnected nodes in the second part represents some hypergraph in the manner described above. Bipartite graph modelĪ hypergraph H may be represented by a bipartite graph BG as follows: the sets X and E are the partitions of BG, and ( x 1, e 1) are connected with an edge if and only if vertex x 1 is contained in edge e 1 in H. The primal graph is sometimes also known as the Gaifman graph of the hypergraph. The primal 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. For a disconnected hypergraph H, G is a host graph if there is a bijection between the connected components of G and of H, such that each connected component G' of G is a host of the corresponding H'.Ī 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 contains at least one vertex from both classes. When a notion of equality is properly defined, as done below, the operation of taking the dual of a hypergraph is an involution, i.e.,Ī 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. The dual of is a hypergraph whose vertices and edges are interchanged, so that the vertices are given by and whose edges are given by where Given a subset, the section hypergraph is the partial hypergraph Given a subset of the index set, the partial hypergraph generated by is the hypergraph The partial hypergraph is a hypergraph with some edges removed. Formally, the subhypergraph induced by a subset of is defined as That is, the vertices are indexed by an index, and the edge set isĪ subhypergraph is a hypergraph with some vertices removed. Let be the hypergraph consisting of vertices The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.īecause hypergraph links can have any cardinality, there are multiple, distinct notions of the concept of a subgraph: subhypergraphs, partial hypergraphs and section hypergraphs. Special cases of hypergraphs include the clutter, where no edge appears as a subset of another edge and the abstract simplicial complex, which contains all subsets of every edge. In cooperative game theory, hypergraphs are called simple games (voting games) this notion is applied to solve problems in social choice theory. In computational geometry, a hypergraph may be called a range space and the hyperedges are called ranges. In particular, there is a Levi graph corresponding to every hypergraph, and vice versa. Hypergraphs can be viewed as incidence structures and vice versa. (In other words, it is a collection of sets of size k.) So a 2-uniform hypergraph is a graph, a 3-uniform hypergraph is a collection of triples, and so on.Ī hypergraph is also called a set system or a family of sets drawn from the universal set X. However, it is often useful 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. While graph edges are pairs of nodes, hyperedges are arbitrary sets of nodes, and can therefore contain an arbitrary number of nodes. Therefore, is a element of, where is the power set of. Formally, a hypergraph is a pair where is a set of elements, called nodes or vertices, and is a set of non-empty subsets of called hyperedges or links. In mathematics, a hypergraph is a generalization of a graph, where an edge can connect any number of vertices. ![]()
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