Results 1  10
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15
Happy Endings for Flip Graphs
, 2007
"... We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include intersections of lattices with convex sets, points on two lines, and several other i ..."
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Cited by 9 (1 self)
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We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include intersections of lattices with convex sets, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.
Graph Drawing with few Slopes
, 2006
"... ... G is the minimum number of distinct edgeslopes in a straightline drawing of G in the plane. We prove that for \Delta> = 5and all large n, there is a \Deltaregular nvertex graph with slopenumber at least n18+"\Delta +4. This is the best known lower bound on the slopenumber of a graphwi ..."
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Cited by 5 (2 self)
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... G is the minimum number of distinct edgeslopes in a straightline drawing of G in the plane. We prove that for \Delta> = 5and all large n, there is a \Deltaregular nvertex graph with slopenumber at least n18+"\Delta +4. This is the best known lower bound on the slopenumber of a graphwith bounded degree. We prove upper and lower bounds on the slopenumberof complete bipartite graphs. We prove a general upper bound on the slopenumber of an arbitrary graph in terms of its bandwidth. It follows that theslopenumber of interval graphs, cocomparability graphs, and ATfree graphs isat most a function of the maximum degree. We prove that graphs of boundeddegree and bounded treewidth have slopenumber at most O(log n). Finallywe prove that every graph has a drawing with one bend per edge, in which thenumber of slopes is at most one more than the maximum degree. In a companionpaper, planar drawings of graphs with few slopes are also considered.
EVERY LARGE POINT SET CONTAINS MANY COLLINEAR POINTS OR AN EMPTY PENTAGON
, 904
"... Abstract. We prove the following generalised empty pentagon theorem: for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique ” conjecture of Kára, Pór ..."
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Cited by 5 (3 self)
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Abstract. We prove the following generalised empty pentagon theorem: for every integer ℓ ≥ 2, every sufficiently large set of points in the plane contains ℓ collinear points or an empty pentagon. As an application, we settle the next open case of the “big line or big clique ” conjecture of Kára, Pór, and Wood [Discrete Comput. Geom. 34(3):497–506, 2005]. 1.
Orthogonal cartograms with few corners per face
, 2010
"... We give an algorithm to create orthogonal drawings of 3connected 3regular planar graphs such that each interior face of the graph is drawn with a prescribed area. This algorithm produces a drawing with at most 12 corners per face and 4 bends per edge, which improves the previous known result of 34 ..."
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Cited by 4 (1 self)
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We give an algorithm to create orthogonal drawings of 3connected 3regular planar graphs such that each interior face of the graph is drawn with a prescribed area. This algorithm produces a drawing with at most 12 corners per face and 4 bends per edge, which improves the previous known result of 34 corners per face.
Private communication
, 2001
"... We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; w ..."
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Cited by 3 (0 self)
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We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; we call a graph with such an embedding an xyz graph. We show that planar xyz graphs can be recognized in linear time, but that it is NPcomplete to determine whether an arbitrary graph is an xyz graph. We also describe an algorithm with running time O(n2 n/2) for testing whether a given nvertex graph is an xyz graph. Submitted:
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
The Topology of Bendless ThreeDimensional Orthogonal Graph Drawing
, 2007
"... We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; ..."
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Cited by 2 (1 self)
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We consider embeddings of 3regular graphs into 3dimensional Cartesian coordinates, in such a way that two vertices are adjacent if and only if two of their three coordinates are equal (that is, if they lie on an axisparallel line) and such that no three points lie on the same axisparallel line; we call a graph with such an embedding an xyz graph. We describe a correspondence between xyz graphs and facecolored embeddings of the graph onto twodimensional manifolds, and we relate bipartiteness of the xyz graph to orientability of the underlying topological surface. Using this correspondence, we show that planar graphs are xyz graphs if and only if they are bipartite, cubic, and threeconnected, and that it is NPcomplete to determine whether an arbitrary graph is an xyz graph. We also describe an algorithm with running time O(n2 n/2) for testing whether a given graph is an xyz graph.
Plane geometric graph augmentation: a generic perspective
, 2011
"... Graph augmentation problems are motivated by network design, and have been studied extensively in optimization. We consider augmentation problems over plane geometric graphs, that is, graphs given with a crossingfree straightline embedding in the plane. The geometric constraints on the possible ne ..."
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Cited by 2 (0 self)
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Graph augmentation problems are motivated by network design, and have been studied extensively in optimization. We consider augmentation problems over plane geometric graphs, that is, graphs given with a crossingfree straightline embedding in the plane. The geometric constraints on the possible new edges render some of the simplest augmentation problems intractable, and in many cases only extremal results are known. We survey recent results, highlight common trends, and gather numerous conjectures and open problems.
A Note on MinimumSegment Drawings of Planar Graphs
"... A straightline drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment. A segment in a straightline drawing is a maximal set of edges that form a straight line segment. A minimumsegment d ..."
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Cited by 1 (1 self)
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A straightline drawing of a planar graph G is a planar drawing of G, where each vertex is mapped to a point on the Euclidean plane and each edge is drawn as a straight line segment. A segment in a straightline drawing is a maximal set of edges that form a straight line segment. A minimumsegment drawing of G is a straightline drawing of G, where the number of segments is the minimum among all possible straightline drawings of G. In this paper we prove that it is NPcomplete to determine whether a plane graph G has a straightline drawing with at most k segments, where k ≥ 3. We also prove that the problem of deciding whether a given partial drawing of G can be extended to a straightline drawing with at most k segments is NPcomplete, even when G is an outerplanar graph. Finally, we investigate a worstcase lower bound on the number of segments required by straightline drawings of arbitrary spanning trees of a given planar graph. 1
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