Squarability of rectangle arrangements

TL;DR

This paper investigates the conditions under which rectangle arrangements can be transformed into square arrangements while preserving structure, providing counterexamples to existing conjectures and introducing a linear program for decision-making.

Contribution

It presents counterexamples to a conjecture about squarability of rectangle arrangements and proposes a linear programming approach for a restricted squarability problem.

Findings

Counterexamples to the conjecture on non-crossing, non-side-piercing arrangements

Counterexamples for transforming box arrangements into hypercube arrangements

A linear program to decide squarability with order preservation

Abstract

We study when an arrangement of axis-aligned rectangles can be transformed into an arrangement of axis-aligned squares in $\mathbb{R}^2$ while preserving its structure. We found a counterexample to the conjecture of J. Klawitter, M. N\"ollenburg and T. Ueckerdt whether all arrangements without crossing and side-piercing can be squared. Our counterexample also works in a more general case when we only need to preserve the intersection graph and we forbid side-piercing between squares. We also show counterexamples for transforming box arrangements into combinatorially equivalent hypercube arrangements. Finally, we introduce a linear program deciding whether an arrangement of rectangles can be squared in a more restrictive version where the order of all sides is preserved.