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

  • Jonathan Klawitter
  • Martin NöllenburgEmail author
  • Torsten Ueckerdt
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9411)


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. Using this, we give a new proof 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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Copyright information

© Springer International Publishing Switzerland 2015

Authors and Affiliations

  • Jonathan Klawitter
    • 1
    • 2
  • Martin Nöllenburg
    • 3
    Email author
  • Torsten Ueckerdt
    • 2
  1. 1.Institut für Theoretische InformatikKarlsruhe Institute of TechnologyKarlsruheGermany
  2. 2.Institut für Algebra und GeometrieKarlsruhe Institute of TechnologyKarlsruheGermany
  3. 3.Algorithms and Complexity GroupTU WienViennaAustria

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