Definition: A hypergraph $\Gamma=(V,\mathcal{E})$ is a set of vertices $V$ and a collection $\mathcal{E}$ of subsets of $V$ such that for every $E\in \mathcal{E}$, we have $|E|\geq 2$. The members of $\mathcal{E}$ are called hypergraphs.
Example: Let $V=\{1,2,3,4\}$ and consider the collection $\mathcal{E}=\{\{1,2,4\}, \{3,4\}, \{2,3,4\}\}$.
However, I have tried to depict it. For instance, $\{1,2,4\}$ means that I've connected 1 and 2, 2 and 4, 1 and 4. I've the following picture.
Is it correct?
2 Answers
$\begingroup$The two main philosophies of hypergraph drawing are:
The Set-Circling School (shown on the left). Here, you just take every hyperedge and circle all the vertices involved with a continuous region.
- Pros: Each edge is definitely unambiguously drawn.
- Cons: Way more going on in the diagram, especially when lots of edges overlap.
The Curved-Line School (shown on the right). Here, you draw a curved line through all the vertices on a hyperedge, in arbitrary order, often sticking out a bit past the ends.
- Pros: Minimalist. A natural generalization of graph drawings.
- Cons: Not always obvious that it's the same edge entering and exiting a vertex.
(Note: in both of these examples, I misread the question and so both of these are drawings of the hypergraph with edge set $\{\{1,2,4\}, \{2,3,4\}, \{2,4\}\}$. Sorry. These drawings took a great deal of effort to make, so I'm not going to draw them all over again.)
In both cases, the problem we are trying to solve is to make a drawing with the single hyperedge $\{1,2,4\}$ look different from from a drawing including some of the hyperedges $\{1,2\}$, $\{1,4\}$, $\{2,4\}$.
$\endgroup$ 2 $\begingroup$A way to see hypergraphs consists of representing its vertices and then enclosing the vertices belonging to a hyperedge into a set.
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