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Article Dans Une Revue Graphs and Combinatorics Année : 2024

The primality graph of critical 3-hypergraphs

Abderrahim Boussaïri
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Brahim Chergui
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Pierre Ille
Mohamed Zaidi
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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.
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Dates et versions

hal-04541203 , version 1 (10-04-2024)

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Abderrahim Boussaïri, Brahim Chergui, Pierre Ille, Mohamed Zaidi. The primality graph of critical 3-hypergraphs. Graphs and Combinatorics, 2024, 40 (3), pp.49. ⟨10.1007/s00373-024-02772-x⟩. ⟨hal-04541203⟩
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