Results 1  10
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21
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 14 (4 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⌋.
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 12 (5 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
, 2007
"... 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 11 (7 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. AMS subject classification: 05C78
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 9 (6 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.
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 4 (2 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.
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 4 (3 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.
SCHNYDER DECOMPOSITIONS FOR REGULAR PLANE GRAPHS AND APPLICATION TO DRAWING
"... Abstract. Schnyder woods are decompositions of simple triangulations into three edgedisjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to dangulations (plane graphs with faces of degree d) for all d ≥ 3. A Schnyder decomposi ..."
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Cited by 3 (2 self)
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Abstract. Schnyder woods are decompositions of simple triangulations into three edgedisjoint spanning trees crossing each other in a specific way. In this article, we define a generalization of Schnyder woods to dangulations (plane graphs with faces of degree d) for all d ≥ 3. A Schnyder decomposition is a set of d spanning forests crossing each other in a specific way, and such that each internal edge is part of exactly d − 2 of the spanning forests. We show that a Schnyder decomposition exists if and only if the girth of the dangulation is d. As in the case of Schnyder woods (d = 3), there are alternative formulationsintermsoforientations (“fractional ” orientations when d ≥ 5)and in terms of cornerlabellings. Moreover, the set of Schnyder decompositions on a fixed dangulation of girth d is a distributive lattice. We also show that the structures dual to Schnyder decompositions (on dregular plane graphs of mincut d rooted at a vertex v ∗ ) are decompositions into d spanning trees rooted atv ∗ such that each edge not incidentto v ∗ isused in opposite directions by two trees. Additionally, for even values of d, we show that a subclass of Schnyder decompositions, which are called even, enjoy additional properties that yield a reduced formulation; in the case d = 4, these correspond to wellstudied structures on simple quadrangulations (2orientations and partitions into 2 spanning trees). In the case d = 4, the dual of even Schnyder decompositions yields (planar) orthogonal and straightline drawing algorithms. For a 4regular plane graph G of mincut 4 with n vertices plus a marked vertex v, the vertices of G\v are placed on a (n−1)×(n−1) grid according to a permutation pattern, and in the orthogonal drawing each of the 2n−2 edges of G\v has exactly one bend. Embedding also the marked vertex v is doable at the cost of two additional rows and columns and 8 additional bends for the 4 edges incident to v. We propose a further compaction step for the drawing algorithm and show that the obtained gridsize is strongly concentrated around 25n/32 ×25n/32 for a uniformly random instance with n vertices. 1.
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 3 (2 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.
A unified bijective method for maps: application to two classes with boundaries
"... boundaries ..."
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 2 (0 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