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34
Boundeddegree graphs have arbitrarily large geometric thickness
, 2008
"... The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 200 ..."
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Cited by 30 (7 self)
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The geometric thickness of a graph G is the minimum integer k such that there is a straight line drawing of G with its edge set partitioned into k plane subgraphs. Eppstein [Separating thickness from geometric thickness. In Towards a Theory of Geometric Graphs, vol. 342 of Contemp. Math., AMS, 2004] asked whether every graph of bounded maximum degree has bounded geometric thickness. We answer this question in the negative, by proving that there exists ∆regular graphs with arbitrarily large geometric thickness. In particular, for all ∆ ≥ 9 and for all large n, there exists a ∆regular graph with geometric thickness at least c √ ∆n 1/2−4/∆−ǫ. Analogous results concerning graph drawings with few edge slopes are also presented, thus solving open problems by Dujmović et al. [Really straight graph drawings. In Proc. 12th
Graph Treewidth and Geometric Thickness Parameters
 DISCRETE AND COMPUTATIONAL GEOMETRY
, 2005
"... Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restri ..."
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Cited by 23 (9 self)
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Consider a drawing of a graph G in the plane such that crossing edges are coloured differently. The minimum number of colours, taken over all drawings of G, is the classical graph parameter thickness. By restricting the edges to be straight, we obtain the geometric thickness. By additionally restricting the vertices to be in convex position, we obtain the book thickness. This paper studies the relationship between these parameters and treewidth. Our first main result states that for graphs of treewidth k, the maximum thickness and the maximum geometric thickness both equal ⌈k/2⌉. This says that the lower bound for thickness can be matched by an upper bound, even in the more restrictive geometric setting. Our second main result states that for graphs of treewidth k, the maximum book thickness equals k if k ≤ 2 and equals k + 1 if k ≥ 3. This refutes a conjecture of Ganley and Heath [Discrete Appl. Math. 109(3):215–221, 2001]. Analogous results are proved for outerthickness, arboricity, and stararboricity.
Really straight graph drawings
 Proc. 12th International Symp. on Graph Drawing (GD ’04
, 2004
"... We study straightline drawings of 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 5n/2 segme ..."
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Cited by 17 (3 self)
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We study straightline drawings of 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 5n/2 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). Drawings of nonplanar graphs with few slopes are also considered. For example, interval graphs, cocomparability graphs and ATfree graphs are shown to have have drawings in which the number of slopes is bounded by the maximum degree. We prove that graphs of bounded degree and bounded treewidth have drawings with O(log n) slopes. Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the
GRAPH DRAWINGS WITH FEW SLOPES
, 2006
"... The slopenumber of a graph G is the minimum number of distinct edge slopes in a straightline drawing of G in the plane. We prove that for ∆ ≥ 5 and all large n, there is a ∆regular nvertex graph with slopenumber at least 8+ε 1− n ∆+4. This is the best known lower bound on the slopenumber of a ..."
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Cited by 16 (3 self)
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The slopenumber of a graph G is the minimum number of distinct edge slopes in a straightline drawing of G in the plane. We prove that for ∆ ≥ 5 and all large n, there is a ∆regular nvertex graph with slopenumber at least 8+ε 1− n ∆+4. This is the best known lower bound on the slopenumber of a graph with bounded degree. We prove upper and lower bounds on the slopenumber of complete bipartite graphs. We prove a general upper bound on the slopenumber of an arbitrary graph in terms of its bandwidth. It follows that the slopenumber of interval graphs, cocomparability graphs, and ATfree graphs is at most a function of the maximum degree. We prove that graphs of bounded degree and bounded treewidth have slopenumber at most O(log n). Finally we prove that every graph has a drawing with one bend per edge, in which the number of slopes is at most one more than the maximum degree. In a companion paper, planar drawings of graphs with few slopes are also considered.
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.
CHARACTERISATIONS AND EXAMPLES OF GRAPH CLASSES WITH BOUNDED EXPANSION
"... Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of t ..."
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Cited by 10 (3 self)
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Classes with bounded expansion, which generalise classes that exclude a topological minor, have recently been introduced by Neˇsetˇril and Ossona de Mendez. These classes are defined by the fact that the maximum average degree of a shallow minor of a graph in the class is bounded by a function of the depth of the shallow minor. Several lineartime algorithms are known for bounded expansion classes (such as subgraph isomorphism testing), and they allow restricted homomorphism dualities, amongst other desirable properties. In this paper we establish two new characterisations of bounded expansion classes, one in terms of socalled topological parameters, the other in terms of controlling dense parts. The latter characterisation is then used to show that the notion of bounded expansion is compatible with ErdösRényi model of random graphs with constant average degree. In particular, we prove that for every fixed d> 0, there exists a class with bounded expansion, such that a random graph of order n and edge probability d/n asymptotically almost surely belongs to the class. We then present several new examples of classes with bounded expansion that do not exclude some topological minor, and appear naturally in the context of graph drawing or graph colouring. In particular, we prove that the following classes have bounded expansion: graphs that can be drawn in the plane with a bounded number of crossings per edge, graphs with bounded stack number, graphs with bounded queue number, and graphs with bounded nonrepetitive chromatic number. We also prove that graphs with ‘linear ’ crossing number are contained in a topologicallyclosed class, while graphs with bounded crossing number are contained in a minorclosed class.
On the Queue Number of Planar Graphs
, 2010
"... We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straightline crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1 ..."
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Cited by 9 (0 self)
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We prove that planar graphs have O(log 4 n) queue number, thus improving upon the previous O ( √ n) upper bound. Consequently, planar graphs admit 3D straightline crossingfree grid drawings in O(n log c n) volume, for some constant c, thus improving upon the previous O(n 3/2) upper bound. 2 1
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.
Partitions of Complete Geometric Graphs into Plane Trees
 IN 12TH INTERNATIONAL SYMPOSIUM ON GRAPH DRAWING (GD ’04
, 2006
"... Consider the open problem: does every complete geometric graph K 2n have a partition of its edge set into n plane spanning trees? We approach this problem from three directions. First, we study the case of convex geometric graphs. It is well known that the complete convex graph K 2n has a partition ..."
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Cited by 7 (2 self)
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Consider the open problem: does every complete geometric graph K 2n have a partition of its edge set into n plane spanning trees? We approach this problem from three directions. First, we study the case of convex geometric graphs. It is well known that the complete convex graph K 2n has a partition into n plane spanning trees. We characterise all such partitions. Second, we give a sufficient condition, which generalises the convex case, for a complete geometric graph to have a partition into plane spanning trees. Finally, we consider a relaxation of the problem in which the trees of the partition are not necessarily spanning. We prove that every complete geometric graph Kn can be partitioned into at most n n/12 plane trees.
Queue Layouts of Graph Products and Powers
, 2005
"... A kqueue layout of a graph G consists of a linear order σ of V (G), and a partition of E(G) into k sets, each of which contains no two edges that are nested in σ. This paper studies queue layouts of graph products and powers. ..."
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Cited by 6 (4 self)
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A kqueue layout of a graph G consists of a linear order σ of V (G), and a partition of E(G) into k sets, each of which contains no two edges that are nested in σ. This paper studies queue layouts of graph products and powers.