What's hypergraph

The definition of Hypergraph has puzzled me for a long time. Today I meet it in the Jung's tutorial, then I ask for the help of Google. Here is a slight digression: you can look up something's definition by entering "Define: xxx" in the search box of Google.

In mathematics, a hypergraph is a generalization of a graph, where edges can connect any number of vertices. Formally, a hypergraph H is a pair H=(X,E) where X is a set of elements, called nodes or vertices, and E is a set of non-empty subsets of X called hyperedges orlinks. Therefore, E is a subset of P(X)\{FI}, where P(X) is the power set of X. While graph edges are pairs of nodes, hyperedges are arbitrary sets of nodes and can therefore contain an arbitrary number of nodes.

A hypergraph is also called a set system or a family of sets drawn from the universal set X. Hypergraphs can be viewed as incidence structures and vice versa. In particular, there is a Levi graph corresponding to every hypergraph, and vice versa.

Unlike graphs, hypergraphs are difficult to draw on paper, so they tned to be studied using the nomenclature of set theory rather than the more pictorial descriptions(like 'trees', 'forests' and 'cycles') of graph theory. Special cases 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.

The collection of hypergraphs is a category with hypergraph homomorphisms as morphisms.

For the more details, please see the original page of wiki: Hypergraph



(The problem is that I am still puzzled by some conception, such as Levi graph, incidence graph and so on.)