The complexity of recognizing $ABAB$-free hypergraphs

TL;DR

The paper establishes that recognizing ABAB-free and ABABA-free hypergraphs is NP-complete for every fixed $k\ge 2$, by reductions from 3-uniform hypergraph 2-colorability using carefully designed gadgets that encode colorings as orderings. It provides a unified inductive framework to extend these hardness results to general $(AB)^k$ and $(AB)^kA$-free families, tying the combinatorial pattern avoidance to geometric realizability questions and the incidence structure of pseudodisks. As a key geometric application, it proves NP-hardness for deciding whether a hypergraph can be realized as the incidence hypergraph of points and pseudodisks. The results illuminate the computational complexity of pattern-avoidance in hypergraphs and highlight important open questions, including the $ABA$-free case and complexity under edge-size constraints.

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.

Theorems & Definitions (21)

• Theorem 1
• Theorem 2
• Theorem 3
• Lemma 5
• proof
• Lemma 6
• proof
• Corollary 8
• proof
• Theorem 10