The complexity of recognizing $ABAB$-free hypergraphs

TL;DR

This paper proves that recognizing $ABAB$-free hypergraphs is NP-complete, extending to similar patterns, and shows that determining if a hypergraph can be realized as points and pseudodisks is also NP-complete.

Contribution

It establishes the NP-completeness of recognizing $ABAB$-free hypergraphs and related patterns, and connects this to the realizability problem for points and pseudodisks.

Findings

Recognition of $ABAB$-free hypergraphs is NP-complete.

Recognition of $ABABA$-free hypergraphs is NP-complete.

Deciding realizability as points and pseudodisks is NP-complete.

Abstract

The study of geometric hypergraphs gave rise to the notion of $ABAB$-free hypergraphs. A hypergraph $\mathcal{H}$ is called $ABAB$-free if there is an ordering of its vertices such that there are no hyperedges $A,B$ and vertices $v_1,v_2,v_3,v_4$ in this order satisfying $v_1,v_3\in A\setminus B$ and $v_2,v_4\in B\setminus A$. In this paper, we prove that it is NP-complete to decide if a hypergraph is $ABAB$-free. We show a number of analogous results for hypergraphs with similar forbidden patterns, such as $ABABA$-free hypergraphs. As an application, we show that deciding whether a hypergraph is realizable as the incidence hypergraph of points and pseudodisks is also NP-complete.