Taxonomic relationship between hypergraph and graph

Question

I am trying to build a taxonomy of graph types. A hypergraph is often defined as a generalization of a graph. Its edges (called hyperedge) can have more then 2 endpoints. From a taxonomic view point, is an hypergraph a superclass of graph or is it a subclass ? If an hypergraph is a subclass of graph, what would be called the category of graphs with only simple edges (2 endpoint edges) ?

Answer 1

You can define a hypergraph as a pair $\mathcal{H}=(V,\mathcal{E})$, where $\mathcal{E}\subseteq 2^{V}$. When all the edges have the same size, say $k$, this is $\mathcal{E}\subseteq\binom{V}{k}$, then $\mathcal{H}$ is called a $k$-uniform hypergraph.

Using the notation from above graphs would correspond to 2-uniform hypergraphs. You can further restrict to graphs not having multiple edges, ie. simple graphs. Two exotic names for these are 2-uniform simple hypergraphs or 2-uniform linear hypergraphs.