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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 20 (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⌋.
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 12 (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.
A binary labelling for plane Laman graphs and quadrangulations
, 2008
"... We present binary labelings for the angles of quadrangulations and plane Laman graphs, which are in analogy with Schnyder labelings for triangulations [W. Schnyder, Proc. 1st ACMSIAM Symposium on Discrete Algorithms, 1990] and imply a special tree decomposition for quadrangulations. In particular, ..."
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Cited by 7 (3 self)
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We present binary labelings for the angles of quadrangulations and plane Laman graphs, which are in analogy with Schnyder labelings for triangulations [W. Schnyder, Proc. 1st ACMSIAM Symposium on Discrete Algorithms, 1990] and imply a special tree decomposition for quadrangulations. In particular, we show how to embed quadrangulations on a 2book, so that each page contains a noncrossing alternating tree.
Straightline drawing of quadrangulations
 In Proceedings of Graph Drawing’06
, 2006
"... Abstract. This article introduces a straightline drawing algorithm for quadrangulations, in the family of the facecounting algorithms. It outputs in linear time a drawing on a regular W ×H grid such that W +H = n − 1 − ∆, where n is the number of vertices and ∆ is an explicit combinatorial paramet ..."
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Cited by 6 (4 self)
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Abstract. This article introduces a straightline drawing algorithm for quadrangulations, in the family of the facecounting algorithms. It outputs in linear time a drawing on a regular W ×H grid such that W +H = n − 1 − ∆, where n is the number of vertices and ∆ is an explicit combinatorial parameter of the quadrangulation. 1
Partitions of graphs into trees
 IN PROCEEDINGS OF GRAPH DRAWING’06 (KARLSRUHE), VOLUME 4372 OF LNCS
, 2007
"... In this paper, we study the ktree partition problem which is a partition of the set of edges of a graph into k edgedisjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) ..."
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Cited by 3 (0 self)
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In this paper, we study the ktree partition problem which is a partition of the set of edges of a graph into k edgedisjoint trees. This problem occurs at several places with applications e.g. in network reliability and graph theory. In graph drawing there is the still unbeaten (n − 2) × (n − 2) area planar straight line drawing of maximal planar graphs using Schnyder’s realizers [15], which are a 3tree partition of the inner edges. Maximal planar bipartite graphs have a 2tree partition, as shown by Ringel [14]. Here we give a different proof of this result with a linear time algorithm. The algorithm makes use of a new ordering which is of interest of its own. Then we establish the NPhardness of the ktree partition problem for general graphs and k ≥ 2. This parallels NPhard partition problems for the vertices [3], but it contrasts the efficient computation of partitions into forests (also known as arboricity) by matroid techniques [7].
4labelings and grid embeddings of plane quadrangulations
 In Proceedings of the 17th International Symposium on Graph Drawing
, 2009
"... We show that each quadrangulation on n vertices has a closed rectangle of influence drawing on the (n − 2) × (n − 2) grid. Further, we present a simple algorithm to obtain a straightline drawing of a quadrangulation on the n ..."
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Cited by 1 (0 self)
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We show that each quadrangulation on n vertices has a closed rectangle of influence drawing on the (n − 2) × (n − 2) grid. Further, we present a simple algorithm to obtain a straightline drawing of a quadrangulation on the n
Lower Bounds on the Area Requirements of SeriesParallel Graphs
, 2009
"... We show that there exist seriesparallel graphs requiring Ω(n2 √ logn) area in any straightline or polyline grid drawing. Such a result is achieved in two steps. First, we show that, in any straightline or polyline drawing of K2,n, one side of the bounding box has length Ω(n), thus answering two ..."
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We show that there exist seriesparallel graphs requiring Ω(n2 √ logn) area in any straightline or polyline grid drawing. Such a result is achieved in two steps. First, we show that, in any straightline or polyline drawing of K2,n, one side of the bounding box has length Ω(n), thus answering two questions posed by Biedl et al. [Information Processing Letters, 2003]. Second, we show a family of seriesparallel graphs requiring Ω(2 √ logn) width and Ω(2 √ logn) height in any straightline or polyline grid drawing. Combining the two results, the Ω(n2 √ logn) area lower bound is achieved. 2 1