Homothetic triangle representations of planar graphs

We prove that every planar graph is the intersection graph of homothetic triangles in the plane.

[1]  Stefan Felsner,et al.  Contact representations of planar graphs with cubes , 2011, SoCG '11.

[2]  E. M. Andreev.  ON CONVEX POLYHEDRA IN LOBAČEVSKIĬ SPACES , 1970 .

[3]  G. G. Stokes.  "J." , 1890, The New Yale Book of Quotations.

[4]  W. Schnyder.  Planar graphs and poset dimension , 1989 .

[5]  Yves Colin de Verdi.  Un principe variationnel pour les empilements de cercles , 1991 .

[6]  L. Lovász.  Geometric representations of graphs , 2014 .

[7]  O. Schramm.  Square tilings with prescribed combinatorics , 1993 .

[8]  Erik Jan van Leeuwen,et al.  Integer Representations of Convex Polygon Intersection Graphs , 2013, SIAM J. Discret. Math..

[9]  Stefan Felsner,et al.  Homothetic Triangle Contact Representations of Planar Graphs , 2007, CCCG.

[10]  Ulrik Brandes,et al.  Graph-Theoretic Concepts in Computer Science , 2002, Lecture Notes in Computer Science.

[11]  Stefan Felsner,et al.  Computing cartograms with optimal complexity , 2012, SoCG '12.

[12]  Michael Kaufmann,et al.  Max-tolerance graphs as intersection graphs: cliques, cycles, and recognition , 2006, SODA '06.

[13]  Stefan Felsner,et al.  Pentagon Contact Representations , 2017, Electron. Notes Discret. Math..

[14]  Patrice Ossona de Mendez,et al.  On Triangle Contact Graphs , 1994, Combinatorics, Probability and Computing.

[15]  Oded Schramm,et al.  Combinatorically Prescribed Packings and Applications to Conformal and Quasiconformal Maps , 2007, 0709.0710.

[16]  Benjamin Lévêque,et al.  Triangle Contact Representations and Duality , 2010, Graph Drawing.

[17]  Daniel Gonçalves,et al.  Dushnik-Miller dimension of TD-Delaunay complexes , 2017, Eur. J. Comb..

[18]  Hendrik Schrezenmaier,et al.  Homothetic Triangle Contact Representations , 2017, WG.

[19]  Stefan Felsner,et al.  Rectangle and Square Representations of Planar Graphs , 2013 .

[20]  Yifan Hu,et al.  On Touching Triangle Graphs , 2010, GD.

[21]  Martin Nöllenburg,et al.  Edge-Weighted Contact Representations of Planar Graphs , 2012, Graph Drawing.

[22]  Stefan Felsner,et al.  Proportional Contact Representations of Planar Graphs , 2012, J. Graph Algorithms Appl..

[23]  Claire Pennarun,et al.  Planar Graphs as L-intersection or L-contact graphs , 2017, SODA.

[24]  Michael Kaufmann,et al.  Optimal Polygonal Representation of Planar Graphs , 2010, LATIN.