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

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

## 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. 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.

## References

1. 1.
Asplund, E., Grünbaum, B.: On a coloring problem. Mathematica Scandinavica 8, 181–188 (1960)
2. 2.
Felsner, S.: Rectangle and square representations of planar graphs. In: Pach, J. (ed.) Thirty Essays in Geometric Graph Theory, pp. 213–248. Springer, New York (2012)Google Scholar
3. 3.
de Fraysseix, H., de Mendez, P.O., Rosenstiehl, P.: On triangle contact graphs. Comb. Probab. Comput. 3, 233–246 (1994)
4. 4.
Fusy, E.: Transversal structures on triangulations: a combinatorial study and straight-line drawings. Discrete Math. 309(7), 1870–1894 (2009)
5. 5.
Imai, H., Asano, T.: Finding the connected components and a maximum clique of an intersection graph of rectangles in the plane. J. Algorithms 4(4), 310–323 (1983)
6. 6.
Kang, R.J., Müller, T.: Arrangements of pseudocircles and circles. Discrete Comput. Geom. 51, 896–925 (2014)
7. 7.
Kant, G., He, X.: Regular edge labeling of $$4$$-connected plane graphs and its applications in graph drawing problems. Theor. Comput. Sci. 172(1–2), 175–193 (1997)
8. 8.
Klawitter, J., Nöllenburg, M., Ueckerdt, T.: Combinatorial properties of triangle-free rectangle arrangements and the squarability problem. CoRR, arXiv:1509.00835, September 2015
9. 9.
Koźmiński, K., Kinnen, E.: Rectangular duals of planar graphs. Networks 15, 145–157 (1985)
10. 10.
Schnyder, W.: Planar graphs and poset dimension. Order 5(4), 323–343 (1989)
11. 11.
Schnyder, W.: Embedding planar graphs on the grid. In: 1st ACM-SIAM Symposium on Discrete Algorithms, SODA 1990, pp. 138–148 (1990)Google Scholar
12. 12.
Thomassen, C.: Interval representations of planar graphs. J. Comb. Theor. Ser. B 40(1), 9–20 (1986)
13. 13.
Ungar, P.: On diagrams representing graphs. J. London Math. Soc. 28, 336–342 (1953)
14. 14.
Yannakakis, M.: The complexity of the partial order dimension problem. SIAM J. Algebraic Discrete Methods 3(3), 351–358 (1982)

© Springer International Publishing Switzerland 2015

## Authors and Affiliations

• Jonathan Klawitter
• 1
• 2
• Martin Nöllenburg
• 3
• 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