Generalization of Schnyder woods to orientable surfaces and applications

Schnyder woods are particularly elegant combinatorial structures with numerous applications concerning planar triangulations and more generally 3-connected planar maps. We propose a simple generalization of Schnyder woods from the plane to maps on orientable surfaces of any genus with a special emphasis on the toroidal case. We provide a natural partition of the set of Schnyder woods of a given map into distributive lattices depending on the surface homology. In the toroidal case we show the existence of particular Schnyder woods with some global properties that are useful for optimal encoding or graph drawing purpose.

[1]  Stefan Felsner,et al.  Lattice Structures from Planar Graphs , 2004, Electron. J. Comb..

[2]  Bojan Mohar,et al.  Tessellation and Visibility Representations of Maps on the Torus , 1998, Discret. Comput. Geom..

[3]  Guillaume Chapuy,et al.  A bijection for covered maps, or a shortcut between Harer-Zagierʼs and Jacksonʼs formulas , 2011, J. Comb. Theory, Ser. A.

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

[5]  David Eppstein,et al.  Steinitz Theorems for Simple Orthogonal Polyhedra , 2014, J. Comput. Geom..

[6]  Dominique Poulalhon,et al.  Optimal Coding and Sampling of Triangulations , 2003, Algorithmica.

[7]  Stefan Felsner,et al.  Geometric Graphs and Arrangements , 2004 .

[8]  Ezra Miller,et al.  Planar graphs as minimal resolutions of trivariate monomial ideals , 2002, Documenta Mathematica.

[9]  Stefan Felsner,et al.  Geodesic Embeddings and Planar Graphs , 2003, Order.

[10]  Torsten Ueckerdt,et al.  Geometric Representations of Graphs with Low Polygonal Complexity , 2012 .

[11]  Nicolas Bonichon,et al.  An Information-Theoretic Upper Bound of Planar Graphs Using Triangulation , 2003, STACS.

[12]  Éric Fusy,et al.  Transversal structures on triangulations: A combinatorial study and straight-line drawings , 2006, Discret. Math..

[13]  David Eppstein,et al.  Drawing Graphs in the Plane with a Prescribed Outer Face and Polynomial Area , 2010, Graph Drawing.

[14]  Paul D. Seymour,et al.  Graph minors. VI. Disjoint paths across a disc , 1986, J. Comb. Theory, Ser. B.

[15]  H. de Fraysseix,et al.  On topological aspects of orientations , 2001, Discret. Math..

[16]  B. Bollobás,et al.  Combinatorics, Probability and Computing , 2006 .

[17]  Stefan Felsner,et al.  Convex Drawings of 3-Connected Plane Graphs , 2005, SODA '05.

[18]  Goos Kant,et al.  Drawing planar graphs using the canonical ordering , 1996, Algorithmica.

[19]  P. Giblin.  Graphs, surfaces, and homology , 1977 .

[20]  Benjamin Lévêque,et al.  Toroidal Maps: Schnyder Woods, Orthogonal Surfaces and Straight-Line Representations , 2014, Discret. Comput. Geom..

[21]  Thomas Lewiner,et al.  Optimal encoding of triangular and quadrangular meshes with fixed topology , 2010, CCCG.

[22]  F. Leighton,et al.  Drawing Planar Graphs Using the Canonical Ordering , 1996 .

[23]  Michael T. Goodrich,et al.  Planar Drawings of Higher-Genus Graphs , 2009, J. Graph Algorithms Appl..

[24]  Carsten Thomassen,et al.  Claw‐decompositions and tutte‐orientations , 2006, J. Graph Theory.

[25]  Nicolas Bonichon,et al.  A bijection between realizers of maximal plane graphs and pairs of non-crossing Dyck paths , 2005, Discret. Math..

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

[27]  Jérémy Barbay,et al.  Succinct Representation of Labeled Graphs , 2007, Algorithmica.

[28]  Benjamin Lévêque,et al.  Encoding Toroidal Triangulations , 2015, Discrete & Computational Geometry.

[29]  Kolja B. Knauer,et al.  Orienting Triangulations , 2016, J. Graph Theory.

[30]  Benjamin Lévêque,et al.  On the structure of Schnyder woods on orientable surfaces , 2015, J. Comput. Geom..

[31]  Benjamin Lévêque,et al.  Triangle Contact Representations and Duality , 2010, Discrete & Computational Geometry.

[32]  Bojan Mohar.  Straight-line representations of maps on the torus and other flat surfaces , 1996, Discret. Math..

[33]  Walter Schnyder,et al.  Embedding planar graphs on the grid , 1990, SODA '90.

[34]  Stefan Felsner,et al.  Posets and planar graphs , 2005, J. Graph Theory.

[35]  Stefan Felsner,et al.  Convex Drawings of Planar Graphs and the Order Dimension of 3-Polytopes , 2001, Order.

[36]  Éric Fusy,et al.  A bijection for triangulations, quadrangulations, pentagulations, etc , 2010, J. Comb. Theory, Ser. A.

[37]  Pierre Rosenstiehl.  Embedding in the Plane With Orientation Constraints: The Angle Graph , 1989 .

[38]  Gilles Schaeer,et al.  Bijective census and random generation of Eulerian planar maps with prescribed vertex degrees , 1997 .

[39]  Thomas Lewiner,et al.  Schnyder Woods for Higher Genus Triangulated Surfaces, with Applications to Encoding , 2009, Discret. Comput. Geom..

[40]  Nicolas Bonichon,et al.  Connections between Theta-Graphs, Delaunay Triangulations, and Orthogonal Surfaces , 2010, WG.

[41]  Lali Barrière,et al.  4-labelings and Grid Embeddings of Plane Quadrangulations , 2012, Discret. Math..

[42]  J. Propp.  Lattice structure for orientations of graphs , 2002, math/0209005.

[43]  Olivier Devillers,et al.  Canonical Ordering for Triangulations on the Cylinder, with Applications to Periodic Straight-Line Drawings , 2012, GD.

[44]  Dominique Poulalhon,et al.  A Generic Method for Bijections between Blossoming Trees and Planar Maps , 2013, Electron. J. Comb..

[45]  Olivier Bernardi,et al.  Bijective Counting of Tree-Rooted Maps and Shuffles of Parenthesis Systems , 2006, Electron. J. Comb..

[46]  Dominique Poulalhon,et al.  Uniform random sampling of simple branched coverings of the sphere by itself , 2014, SODA.

[47]  Stefan Felsner,et al.  Schnyder Woods and Orthogonal Surfaces , 2006, GD.