[Paper Review] Circular-arc hypergraphs: Rigidity via Connectedness

Johannes Köbler, Sebastian Kuhnert+1 more|arXiv (Cornell University)|Dec 4, 2013

Advanced Graph Theory Research4 citations

TL;DR

This paper establishes conditions under which circular-arc hypergraphs and their neighborhood hypergraphs have a unique arc ordering up to reversal, using strict connectedness and tight ordering properties. The key result shows that for twin-free, connected proper circular-arc graphs with non-bipartite or connected complements, the closed neighborhood hypergraph has a unique tight arc ordering, implying essentially unique intersection representations.

ABSTRACT

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 $A$ is a nonempty subset of $B$ and $B$ is not equal to $V(H)$, 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, then $N[G]$ has, up to reversing, a unique arc ordering. If the complement of $G$ 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 based on a theory of local tournaments.

Motivation & Objective

• To characterize when circular-arc hypergraphs have a unique arc ordering up to reversal.
• To define and analyze tight arc orderings as a stronger form of ordering with endpoint-sharing constraints for nested hyperedges.
• To establish conditions under which the closed neighborhood hypergraph of a proper circular-arc graph has a unique tight arc ordering.
• To connect uniqueness of arc orderings to the structure of the graph’s complement, particularly whether it is bipartite or connected.
• To unify results from prior work on local tournaments and intersection representations via orientation theory.

Proposed method

• Introduce the concept of strictly overlap-connected hypergraphs, where hyperedges strictly overlap and the hypergraph is connected under the strict overlap relation.
• Define a tight arc ordering as one where every pair of nested hyperedges $ A \subset B $ (with $ B \neq V $) share a common endpoint in the circular ordering.
• Prove that twin-free, strictly overlap-connected CA hypergraphs have a unique arc ordering up to reversal.
• Use the closed neighborhood hypergraph $ \mathcal{N}[G] $ of a graph $ G $ to characterize proper circular-arc graphs via tight arc orderings.
• Relate arc orderings to orientations: show that a proper arc representation induces a round orientation, and vice versa.
• Leverage Roberts’ uniqueness theorem and results from Deng, Hell, and Huang to derive uniqueness of round orientations from uniqueness of arc representations.

Experimental results

Research questions

• RQ1 Under what conditions does a circular-arc hypergraph have a unique arc ordering up to reversal?
• RQ2 How do strict connectedness and tightness conditions affect the uniqueness of arc orderings in hypergraphs?
• RQ3 What structural properties of a proper circular-arc graph ensure uniqueness of its intersection representation?
• RQ4 How is the uniqueness of arc orderings in the closed neighborhood hypergraph $ \mathcal{N}[G] $ related to the complement $ \overline{G} $ of the graph $ G $?
• RQ5 To what extent do round orientations of a graph correspond to unique arc representations, and how does this relate to the graph’s complement?

Key findings

• A twin-free, strictly overlap-connected circular-arc hypergraph has, up to reversal, a unique arc ordering.
• For a connected, twin-free proper circular-arc graph $ G $, if $ \overline{G} $ is non-bipartite or connected, then $ \mathcal{N}[G] $ has, up to reversal, a unique tight arc ordering.
• If $ \overline{G} $ is bipartite and connected, then $ \mathcal{N}[G] $ has, up to reversal, exactly two tight arc orderings.
• As a corollary, such graphs have an essentially unique intersection representation via proper arc representations.
• The uniqueness of arc orderings in $ \mathcal{N}[G] $ corresponds to the uniqueness of round orientations in $ G $, up to reversal.
• The results are equivalent to those of Deng, Hell, and Huang on unique round orientations, but derived via geometric and hypergraph-theoretic methods rather than tournament theory.