The primality graph of critical 3-hypergraphs
Résumé
Given a 3-hypergraph $H$,
a subset $M$ of $V(H)$ is a module of $H$ if for each $e\in E(H)$ such that
$e\cap M\neq\emptyset$ and
$e\setminus M\neq\emptyset$, there exists $m\in M$ such that $e\cap M=\{m\}$ and for every $n\in M$, we have
$(e\setminus\{m\})\cup\{n\}\in E(H)$.
For example,
$\emptyset$, $V(H)$ and $\{v\}$, where $v\in V(H)$, are modules of $H$, called trivial.
A 3-hypergraph is prime if all its modules are trivial.
Furthermore, a prime 3-hypergraph is critical if all its induced subhypergraphs, obtained by removing one vertex, are not prime.
Lastly, we associate with a prime 3-hypergraph its primality graph the edges of which are the unordered pairs of vertices whose removal provides a prime induced subhypergraph.
We characterize the critical 3-hypergraphs together with their primality graph.