A Note on Saturation for Berge-G Hypergraphs

Abstract

For a graph $$G=(V,E)$$ , a hypergraph H is called Berge-G if there is a hypergraph $$H'$$ , isomorphic to H, so that $$V(G)\subseteq V(H')$$ and there is a bijection $$\phi : E(G) \rightarrow E(H')$$ such that for each $$e\in E(G)$$ , $$e \subseteq \phi (e)$$ . The set of all Berge-G hypergraphs is denoted $$\mathcal {B}(G)$$ . A hypergraph H is called Berge-Gsaturated if it does not contain any subhypergraph from $$\mathcal {B}(G)$$ , but adding any new hyperedge of size at least 2 to H creates such a subhypergraph. Since each Berge-G hypergraph contains |E(G)| hypergedges, it follows that each Berge-G saturated hypergraph must have at least $$|E(G)|-1$$ edges. We show that for each graph G that is not a certain star and for any $$n\ge |V(G)|$$ , there are Berge-G saturated hypergraphs on n vertices and exactly $$|E(G)|-1$$ hyperedges. This solves a problem of finding a saturated hypergraph with the smallest number of edges exactly.