$e \in E$ are of the form $e = \{v, w\}$ with $v, w \in V$; in this case the graph is undirected and the hyperedges are called *edges*. We call a hypergraph $G = (V, E)$ *finite* if $V$ is a finite set, in which case $|E| \leq 2^{|V|}$, so $E$ is finite as well.

![Hypergraph $G = (V, E)$ with $V = \{1, 2, 3, 4, 5\}$ and $E = \{\{1\}, \{1, 2\}, \{2, 3, 4\}, \{3, 4\}\}$.[]{data-label="fig:hgexample"}](hg0){width="3.5cm"}

Hypergraphs possess a natural notion of *duality*, where the roles played by the vertices and hyperedges are interchanged.

\[def:hypergraphdual\] Let $G = (V, E)$ be a hypergraph.

Then we define its *dual hypergraph* as $G^* := (V^*, E^*)$ where $V^* := E$ and $$E^* := \{ \{e \in E {\ |\ }e \ni v \} {\ |\ }v \in V\}.$$ If the hypergraph is weighted, we also exchange the vertex weights and hyperedge costs to make its dual a weighted hypergraph.

The dual of the dual of a hypergraph is isomorphic to the original hypergraph, $(G^*)^* \cong G$.

  **Name**                                      $\mathbf{V}$                                                                           $\mathbf{E}$
  ------------------------------ ------------------------------------------- ---------------------------------------------------------------------------------------------------------------------------------
  Symmetric [@Parter1961]                    $\{1, \ldots, m\}$                                         $\{ \{i, j\} {\ |\ }1 \leq i \leq m, 1 