Representing Hypergraphs by Point-Line Incidences

TL;DR

This paper investigates hypergraph visualization methods using points and curves, analyzing various constraints and complexities, including hardness results and polynomial algorithms, and explores the impact of allowing bends in the curves.

Contribution

It provides a comprehensive complexity analysis of hypergraph representations with points and curves, including hardness proofs and polynomial-time algorithms for specific cases, and discusses bend allowances.

Findings

Six of eight decision variants are $orall ext{R}$-hard.

Polynomial algorithms exist for some restricted cases.

Allowing bends in curves generalizes previous counterexamples.

Abstract

We consider hypergraph visualizations that represent vertices as points in the plane and hyperedges as curves passing through the points of their incident vertices. Specifically, we consider several different variants of this problem by (a) restricting the curves to be lines or line segments, (b) allowing two curves to cross if they do not share an element, or not; and (c) allowing two curves to overlap or not. We show $\exists\mathbb{R}$-hardness for six of the eight resulting decision problem variants and describe polynomial-time algorithms in some restricted settings. Lastly, we briefly touch on what happens if we allow the lines of the represented hyperedges to have bends - to this we generalize a counterexample to a long-standing result that was sometimes assumed to be correct.