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.