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Drawings of planar graphs with few slopes and segments
 Computational Geometry Theory and Applications 38:194–212
, 2005
"... 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 18 (5 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 2n 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.
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 14 (8 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.
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 6 (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.
Drawing cubic graphs with at most five slopes
"... Abstract. We show that every graph G with maximum degree three has a straightline drawing in the plane using edges of at most five different slopes. Moreover, if every connected component of G has at least one vertex of degree less than three, then four directions suffice. 1 ..."
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Cited by 2 (1 self)
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Abstract. We show that every graph G with maximum degree three has a straightline drawing in the plane using edges of at most five different slopes. Moreover, if every connected component of G has at least one vertex of degree less than three, then four directions suffice. 1
Boundeddegree graphs have arbitrarily large geometric thickness
 Electron. J. Combin
, 509
"... It is proved that there exist graphs of bounded degree with arbitrarily large queuenumber. In particular, for all ∆ ≥ 3 and for all sufficiently large n, there is a simple ∆regular nvertex graph with queuenumber at least c √ ∆n 1/2−1/∆ for some absolute constant c. ..."
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Cited by 1 (1 self)
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It is proved that there exist graphs of bounded degree with arbitrarily large queuenumber. In particular, for all ∆ ≥ 3 and for all sufficiently large n, there is a simple ∆regular nvertex graph with queuenumber at least c √ ∆n 1/2−1/∆ for some absolute constant c.
DISTINCT DISTANCES IN GRAPH DRAWINGS
, 2008
"... The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite g ..."
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Cited by 1 (0 self)
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The distancenumber of a graph G is the minimum number of distinct edgelengths over all straightline drawings of G in the plane. This definition generalises many wellknown concepts in combinatorial geometry. We consider the distancenumber of trees, graphs with no K − 4minor, complete bipartite graphs, complete graphs, and cartesian products. Our main results concern the distancenumber of graphs with bounded degree. We prove that nvertex graphs with bounded maximum degree and bounded treewidth have distancenumber in O(log n). To conclude such a logarithmic upper bound, both the degree and the treewidth need to be bounded. In particular, we construct graphs with treewidth 2 and polynomial distancenumber. Similarly, we prove that there exist graphs with maximum degree 5 and arbitrarily large distancenumber. Moreover, as ∆ increases the existential lower bound on the distancenumber of ∆regular graphs tends to Ω(n0.864138). 1
Cubic graphs have bounded slope parameter
, 2009
"... We show that every finite connected graph G with maximum degree three and with at least one vertex of degree smaller than three has a straightline drawing in the plane satisfying the following conditions. No three vertices are collinear, and a pair of vertices form an edge in G if and only if the s ..."
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We show that every finite connected graph G with maximum degree three and with at least one vertex of degree smaller than three has a straightline drawing in the plane satisfying the following conditions. No three vertices are collinear, and a pair of vertices form an edge in G if and only if the segment connecting them is parallel to one of the sides of a previously fixed regular pentagon. It is also proved that every finite graph with maximum degree three permits a straightline drawing with the above properties using at most seven different edge slopes. Submitted:
On Graph Thickness, Geometric Thickness, and Separator Theorems
"... We investigate the relationship between geometric thickness and the thickness, outerthickness, and arboricity of graphs. In particular, we prove that all graphs with arboricity two or outerthickness two have geometric thickness O(log n). The technique used can be extended to other classes of graphs ..."
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We investigate the relationship between geometric thickness and the thickness, outerthickness, and arboricity of graphs. In particular, we prove that all graphs with arboricity two or outerthickness two have geometric thickness O(log n). The technique used can be extended to other classes of graphs so long as a standard separator theorem exists. For example, we can apply it to show the known bound that thickness two graphs have geometric thickness O ( √ n), yielding a simple construction in the process. 1
Testing Bipartiteness of Geometric Intersection Graphs
, 2007
"... We show how to test whether an intersection graph of n line segments or simple polygons in the plane, or of balls in R d, is bipartite, in time O(n log n). More generally we find subquadratic algorithms for connectivity and bipartiteness testing of intersection graphs of a broad class of geometric o ..."
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We show how to test whether an intersection graph of n line segments or simple polygons in the plane, or of balls in R d, is bipartite, in time O(n log n). More generally we find subquadratic algorithms for connectivity and bipartiteness testing of intersection graphs of a broad class of geometric objects. Our algorithms for these problems return either a bipartition of the input or an odd cycle in its intersection graph. We also consider lower bounds for connectivity and kcolorability problems of geometric intersection graphs. For unit balls in R d, connectivity testing has equivalent randomized complexity to construction of Euclidean minimum spanning trees, and for line segments in the plane connectivity testing has the same lower bounds as Hopcroft’s pointline incidence testing problem; therefore, for these problems, connectivity is unlikely to be solved as efficiently as bipartiteness. For line segments or planar disks, testing kcolorability of intersection graphs for k> 2 is NPcomplete. 1