Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem

Jonathan Klawitter; Martin Nöllenburg; Torsten Ueckerdt

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arXiv · 1509.00835

Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem

Jonathan Klawitter·Martin Nöllenburg·Torsten Ueckerdt

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Abstract

We consider arrangements of axis-aligned rectangles in the plane. A geometric arrangement specifies the coordinates of all rectangles, while a combinatorial arrangement specifies only the respective intersection type in which each pair of rectangles intersects. First, we investigate combinatorial contact arrangements, i.e., arrangements of interior-disjoint rectangles, with a triangle-free intersection graph. We show that such rectangle arrangements are in bijection with the 4-orientations of an underlying planar multigraph and prove that there is a corresponding geometric rectangle contact arrangement. Moreover, we prove that every triangle-free planar graph is the contact graph of such an arrangement. Secondly, we introduce the question whether a given rectangle arrangement has a combinatorially equivalent square arrangement. In addition to some necessary conditions and counterexamples, we show that rectangle arrangements pierced by a horizontal line are squarable under certain sufficient conditions.

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Jonathan Klawitter, Martin Nöllenburg, Torsten Ueckerdt. 2015-09-02. Combinatorial Properties of Triangle-Free Rectangle Arrangements and the Squarability Problem. https://arxiv.org/abs/1509.00835

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