Circular-arc hypergraphs: Rigidity via Connectedness

A circular-arc hypergraph H is a hypergraph admitting an arc ordering, that is, a circular ordering of the vertex set V (H) such that every hyperedge is an arc of consecutive vertices. An arc ordering is tight if, for any two hyperedges A and B such that ∅ 6 = A ⊆ B 6= V (H), the corresponding arcs share a common endpoint. We give sufficient conditions for H to have, up to reversing, a unique arc ordering and a unique tight arc ordering. These conditions are stated in terms of connectedness properties of H. It is known that G is a proper circular-arc graph exactly when its closed neighborhood hypergraph N [G] admits a tight arc ordering. We explore connectedness properties of N [G] and prove that, if G is a connected, twin-free, proper circular-arc graph with non-bipartite complement G, then N [G] has, up to reversing, a unique arc ordering. If G is bipartite and connected, then N [G] has, up to reversing, two tight arc orderings. As a corollary, we notice that in both of the two cases G has an essentially unique intersection representation. The last result also follows from the work by Deng, Hell, and Huang [4] based on a theory of local tournaments.