The primality graph of critical 3-hypergraphs

Abstract

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 $e\cap M\ne \emptyset$ and $e{\setminus } M\ne \emptyset$, 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.