Critical 3-hypergraphs (detailed version)

TL;DR

This paper characterizes critical 3-hypergraphs, a class of hypergraphs where removing any vertex results in a non-prime hypergraph, expanding understanding of their structure and properties.

Contribution

It provides a complete characterization of critical 3-hypergraphs, a class previously not fully understood, based on the properties of modules and primeness.

Findings

Critical 3-hypergraphs are fully characterized.

Critical 3-hypergraphs have specific structural properties.

The results extend the theory of hypergraph modules and primeness.

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 $e\cap M\neq\emptyset$ 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. We characterize the critical 3-hypergraphs.