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21
On Linear Layouts of Graphs
, 2004
"... In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp... ..."
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Cited by 33 (22 self)
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In a total order of the vertices of a graph, two edges with no endpoint in common can be crossing, nested, or disjoint. A kstack (resp...
Drawings of planar graphs with few slopes and segments
, 2007
"... We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5 2 ..."
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Cited by 25 (6 self)
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We study straightline drawings of planar graphs with few segments and few slopes. Optimal results are obtained for all trees. Tight bounds are obtained for outerplanar graphs, 2trees, and planar 3trees. We prove that every 3connected plane graph on n vertices has a plane drawing with at most 5 2 n segments and at most 2n slopes. We prove that every cubic 3connected plane graph has a plane drawing with three slopes (and three bends on the outerface). In a companion paper, drawings of nonplanar graphs with few slopes are also considered.
Bijections for Baxter Families and Related Objects
, 2008
"... The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2) ..."
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Cited by 20 (8 self)
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The Baxter number Bn can be written as Bn = � n 0 Θk,n−k−1 with Θk,ℓ = 2 (k + 1) 2 (k + 2)
Rectangle and Square Representations of Planar Graphs
"... In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the recta ..."
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Cited by 18 (6 self)
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In the first part of this survey we consider planar graphs that can be represented by a dissections of a rectangle into rectangles. In rectangular drawings the corners of the rectangles represent the vertices. The graph obtained by taking the rectangles as vertices and contacts as edges is the rectangular dual. In visibility graphs and segment contact graphs the vertices correspond to horizontal or to horizontal and vertical segments of the dissection. Special orientations of graphs turn out to be helpful when dealing with characterization and representation questions. Therefore, we look at orientations with prescribed degrees, bipolar orientations, separating decompositions, and transversal structures. In the second part we ask for representations by a dissections of a rectangle into squares. We
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.
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
Orthogonal Segment Stabbing
, 2004
"... We study a class of geometric stabbing/covering problems for sets of line segments, rays, and lines in the plane. While we demonstrate that the problems on sets of horizontal/vertical line segments are NPcomplete, we show that versions involving (parallel) rays or lines are polynomially solvable. ..."
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Cited by 6 (0 self)
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We study a class of geometric stabbing/covering problems for sets of line segments, rays, and lines in the plane. While we demonstrate that the problems on sets of horizontal/vertical line segments are NPcomplete, we show that versions involving (parallel) rays or lines are polynomially solvable.