In mathematics, a hypergraph is a generalization of a graph in which an edge can join any number of vertices. In contrast, in an ordinary graph, an edge connects exactly two vertices.

Formally, a directed hypergraph is a pair ${\displaystyle (X,E)}$, where ${\displaystyle X}$ is a set of elements called nodes, vertices, points, or elements and ${\displaystyle E}$ is a set of pairs of subsets of ${\displaystyle X}$. Each of these pairs ${\displaystyle (D,C)\in E}$ is called an edge or hyperedge; the vertex subset ${\displaystyle D}$ is known as its tail or domain, and ${\displaystyle C}$ as its head or codomain.

The order of a hypergraph ${\displaystyle (X,E)}$ is the number of vertices in ${\displaystyle X}$. The size of the hypergraph is the number of edges in ${\displaystyle E}$. The order of an edge ${\displaystyle e=(D,C)}$ in a directed hypergraph is ${\displaystyle |e|=(|D|,|C|)}$: that is, the number of vertices in its tail followed by the number of vertices in its head.

The definition above generalizes from a directed graph to a directed hypergraph by defining the head or tail of each edge as a set of vertices (${\displaystyle C\subseteq X}$ or ${\displaystyle D\subseteq X}$) rather than as a single vertex. A graph is then the special case where each of these sets contains only one element. Hence any standard graph theoretic concept that is independent of the edge orders ${\displaystyle |e|}$ will generalize to hypergraph theory.

Under one definition, an undirected hypergraph ${\displaystyle (X,E)}$ is a directed hypergraph which has a symmetric edge set: If ${\displaystyle (D,C)\in E}$ then ${\displaystyle (C,D)\in E}$. For notational simplicity one can remove the "duplicate" hyperedges since the modifier "undirected" is precisely informing us that they exist: If $$Zdroj:https://en.wikipedia.org?pojem=Hypergraph
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