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29
Transversal structures on triangulations, combinatorial study and straightline drawing
, 2007
"... This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edgelabelling and consists of two transversal bip ..."
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Cited by 25 (5 self)
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This article focuses on a combinatorial structure specific to triangulated plane graphs with quadrangular outer face and no separating triangle, called irreducible triangulations. The structure has been introduced by Xin He under the name of regular edgelabelling and consists of two transversal bipolar orientations. For this reason, the terminology used here is that of transversal structures. The main results obtained in the article are a bijection between irreducible triangulations and ternary trees, and a straightline drawing algorithm for irreducible triangulations. For a random irreducible triangulation with n vertices, the grid size of the drawing is asymptotically with high probability 11n/27 × 11n/27 up to an additive error of O ( √ n). In contrast, the best previously known algorithm for these triangulations only guarantees a grid size (⌈n/2 ⌉ − 1) × ⌊n/2⌋.
Tutte Polynomial, Subgraphs Orientations, and Sandpile Model: New Connections via Embeddings
 Electronic J. of Combinatorics
"... We define a bijection between spanning subgraphs and orientations of graphs and explore its enumerative consequences regarding the Tutte polynomial. We obtain unifying bijective proofs for all the evaluations TG(i, j), 0 ≤ i, j ≤ 2 of the Tutte polynomial in terms of subgraphs, orientations, outdegr ..."
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Cited by 16 (1 self)
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We define a bijection between spanning subgraphs and orientations of graphs and explore its enumerative consequences regarding the Tutte polynomial. We obtain unifying bijective proofs for all the evaluations TG(i, j), 0 ≤ i, j ≤ 2 of the Tutte polynomial in terms of subgraphs, orientations, outdegree sequences and sandpile configurations. For instance, for any graph G, we obtain a bijection between connected subgraphs (counted by TG(1, 2)) and rootconnected orientations, a bijection between forests (counted by TG(2, 1)) and outdegree sequences and bijections between spanning trees (counted by TG(1, 1)), rootconnected outdegree sequences and recurrent sandpile configurations. All our proofs are based on a single bijection Φ between the spanning subgraphs and the orientations that we specialize in various ways. The bijection Φ is closely related to a recent characterization of the Tutte polynomial relying on combinatorial embeddings of graphs, that is, on a choice of cyclic order of the edges around each vertex. 1
Transversal structures on triangulations, with application to straight line drawing
 LECTURE NOTES IN COMPUTER SCIENCE
, 2005
"... We define and study a structure called transversal edgepartition related to triangulations without non empty triangles, which is equivalent to the regular edge labeling discovered by Kant and He. We study other properties of this structure and show that it gives rise to a new straightline drawing ..."
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Cited by 14 (6 self)
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We define and study a structure called transversal edgepartition related to triangulations without non empty triangles, which is equivalent to the regular edge labeling discovered by Kant and He. We study other properties of this structure and show that it gives rise to a new straightline drawing algorithm for triangulations without non empty triangles, and more generally for 4connected plane graphs with at least 4 border vertices. Taking uniformly at random such a triangulation with 4 border vertices and n vertices, the size of the grid is almost surely n
Binary labelings for plane quadrangulations and their relatives
, 2008
"... Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder’s one for tr ..."
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Cited by 13 (8 self)
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Motivated by the bijection between Schnyder labelings of a plane triangulation and partitions of its inner edges into three trees, we look for binary labelings for quadrangulations (whose edges can be partitioned into two trees). Our labeling resembles many of the properties of Schnyder’s one for triangulations: Apart from being in bijection with tree decompositions, paths in these trees allow to define the regions of a vertex such that counting faces in them yields an algorithm for embedding the quadrangulation, in this case on a 2book. Furthermore, as Schnyder labelings have been extended to 3connected plane graphs, we are able to extend our labeling from quadrangulations to a larger class of 2connected bipartite graphs. Finally, we propose a binary labeling for Laman graphs.
Geodesic Embeddings and Planar Graphs
, 2002
"... Schnyder labelings are known to have close links to order dimension and drawings of planar graphs. It was observed by Ezra Miller that geodesic embeddings of planar graphs are another class of combinatorial or geometric objects closely linked to Schnyder labelings. We aim to contribute to a better u ..."
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Cited by 13 (8 self)
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Schnyder labelings are known to have close links to order dimension and drawings of planar graphs. It was observed by Ezra Miller that geodesic embeddings of planar graphs are another class of combinatorial or geometric objects closely linked to Schnyder labelings. We aim to contribute to a better understanding of the connections between these objects. In this article we prove a characterization of 3connected planar graphs as those graphs admitting rigid geodesic embeddings, a bijection between Schnyder labelings and rigid geodesic embeddings, a strong version of the BrightwellTrotter theorem.
A bijection for triangulations, for quadrangulations, for pentagulations, etc
"... A dangulation is a planar map with faces of degree d. We present for each integer d ≥ 3 a bijection between the class of dangulations of girth d and a class of decorated plane trees. Each of the bijections is obtained by specializing a “master bijection ” which extends an earlier construction of t ..."
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Cited by 11 (6 self)
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A dangulation is a planar map with faces of degree d. We present for each integer d ≥ 3 a bijection between the class of dangulations of girth d and a class of decorated plane trees. Each of the bijections is obtained by specializing a “master bijection ” which extends an earlier construction of the first author. Bijections already existed for triangulations (d = 3) and for quadrangulations (d = 4). As a matter of fact, our construction unifies a bijection by Fusy, Poulalhon and Schaeffer for triangulations and a bijection by Schaeffer for quadrangulations. For d ≥ 5, both the bijections and the enumerative results are new. We also extend our bijections so as to enumerate pgonal dangulations, that is, dangulations with a simple boundary of length p. We thereby recover bijectively the results of Brown for pgonal triangulations and quadrangulations and establish new results for d ≥ 5. A key ingredient in our proofs is a class of orientations characterizing dangulations of girth d. Earlier results by Schnyder and by De Fraisseyx and Ossona de Mendez showed that triangulations of girth 3 and quadrangulations of girth 4 are characterized by the existence of orientations having respectively indegree 3 and 2 at each inner vertex. We extend this characterization by showing that a dangulation has girth d if and only if the graph obtained by duplicating each edge d − 2 times admits an orientation having indegree d at each inner vertex.
On the number of planar orientations with prescribed degrees
, 2008
"... We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with outdegrees prescribed by a function α: V → N unifies many diffe ..."
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Cited by 9 (3 self)
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We deal with the asymptotic enumeration of combinatorial structures on planar maps. Prominent instances of such problems are the enumeration of spanning trees, bipartite perfect matchings, and ice models. The notion of orientations with outdegrees prescribed by a function α: V → N unifies many different combinatorial structures, including the afore mentioned. We call these orientations αorientations. The main focus of this paper are bounds for the maximum number of αorientations that a planar map with n vertices can have, for different instances of α. We give examples of triangulations with 2.37 n Schnyder woods, 3connected planar maps with 3.209 n Schnyder woods and inner triangulations with 2.91 n bipolar orientations. These lower bounds are accompanied by upper bounds of 3.56 n, 8 n and 3.97 n respectively. We also show that for any planar map M and any α the number of αorientations is bounded from above by 3.73 n and describe a family of maps which have at least 2.598 n αorientations.
Combinatorial and Geometric Properties of Planar Laman Graphs
"... Abstract. Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar barandjoint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial struct ..."
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Cited by 7 (3 self)
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Abstract. Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar barandjoint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial structures for planar Laman graphs: angular structures, angle labelings, and edge labelings. The latter two structures are related to Schnyder realizers for maximally planar graphs. We prove that planar Laman graphs are exactly the class of graphs that have an angular structure that is a tree, called angular tree, and that every angular tree has a corresponding angle labeling and edge labeling. Using a combination of these powerful combinatorial structures, we show that every planar Laman graph has an Lcontact representation, that is, planar Laman graphs are contact graphs of axisaligned Lshapes. Moreover, we show that planar Laman graphs and their subgraphs are the only graphs that can be represented this way. We present efficient algorithms that compute, for every planar Laman graph G, an angular tree, angle labeling, edge labeling, and finally an Lcontact representation of G. The overall running time is O(n 2), where n is the number of vertices of G, and the Lcontact representation is realized on the n × n grid.
Generic method for bijections between blossoming trees and planar maps
 Electron. J. Combin
"... This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling halfedges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by lo ..."
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Cited by 6 (2 self)
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This article presents a unified bijective scheme between planar maps and blossoming trees, where a blossoming tree is defined as a spanning tree of the map decorated with some dangling halfedges that enable to reconstruct its faces. Our method generalizes a previous construction of Bernardi by loosening its conditions of applications so as to include annular maps, that is maps embedded in the plane with a root face different from the outer face. The bijective construction presented here relies deeply on the theory of αorientations introduced by Felsner, and in particular on the existence of minimal and accessible orientations. Since most of the families of maps can be characterized by such orientations, our generic bijective method is proved to capture as special cases all previously known bijections involving blossoming trees: for example Eulerian maps, mEulerian maps, non separable maps and simple triangulations and quadrangulations of a kgon. Moreover, it also permits to obtain new bijective constructions for bipolar orientations and dangulations of girth d of a kgon. As for applications, each specialization of the construction translates into enumerative byproducts, either via a closed formula or via a recursive computational scheme. Besides, for every family of maps described in the paper, the construction can be implemented in linear time. It yields thus an effective way to encode and generate planar maps. In a recent work, Bernardi and Fusy introduced another unified bijective scheme, we adopt here a different strategy which allows us to capture different bijections. These two approaches should be seen as two complementary ways of unifying bijections between planar maps and decorated trees.
Schnyder Woods for Higher Genus Triangulated Surfaces
 SCG'08
, 2008
"... Schnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertexspanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere ..."
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Cited by 5 (1 self)
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Schnyder woods are a well known combinatorial structure for planar graphs, which yields a decomposition into 3 vertexspanning trees. Our goal is to extend definitions and algorithms for Schnyder woods designed for planar graphs (corresponding to combinatorial surfaces with the topology of the sphere, i.e., of genus 0) to the more general case of graphs embedded on surfaces of arbitrary genus. First, we define a new traversal order of the vertices of a triangulated surface of genus g together with an orientation and coloration of the edges that extends the one proposed by Schnyder for the planar case. As a byproduct we show how some recent schemes for compression and compact encoding of graphs can be extended to higher genus. All the algorithms presented here have linear time complexity.