Results 1  10
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12
Flips in Planar Graphs
, 2006
"... We review a selection of results concerning edge flips in triangulations and planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another. We st ..."
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Cited by 29 (5 self)
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We review a selection of results concerning edge flips in triangulations and planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another. We study the problem both from a combinatorial perspective (where only a combinatorial embedding of the graph is specified) and a geometric perspective (where the graph is embedded in the plane, vertices are points and edges are straightline segments). We highlight both the similarities and differences of the two settings, describe many extensions and generalizations, outline several applications and mention open problems.
Planar decompositions and the crossing number of graphs with an excluded minor
 IN GRAPH DRAWING 2006; LECTURE NOTES IN COMPUTER SCIENCE 4372
, 2007
"... Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar ..."
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Cited by 14 (1 self)
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Tree decompositions of graphs are of fundamental importance in structural and algorithmic graph theory. Planar decompositions generalise tree decompositions by allowing an arbitrary planar graph to index the decomposition. We prove that every graph that excludes a fixed graph as a minor has a planar decomposition with bounded width and a linear number of bags. The crossing number of a graph is the minimum number of crossings in a drawing of the graph in the plane. We prove that planar decompositions are intimately related to the crossing number. In particular, a graph with bounded degree has linear crossing number if and only if it has a planar decomposition with bounded width and linear order. It follows from the above result about planar decompositions that every graph with bounded degree and an excluded minor has linear crossing number. Analogous results are proved for the convex and rectilinear crossing numbers. In particular, every graph with bounded degree and bounded treewidth has linear convex crossing number, and every K3,3minorfree graph with bounded degree has linear rectilinear crossing number.
Reconfigurations of polygonal structures
, 2005
"... This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygona ..."
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Cited by 6 (2 self)
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This thesis contains new results on the subject of polygonal structure reconfiguration. Specifically, the types of structures considered here are polygons, polygonal chains, triangulations, and polyhedral surfaces. A sequence of vertices (points), successively joined by straight edges, is a polygonal chain. If the sequence is cyclic, then the object is a polygon. A planar triangulation is a set of vertices with a maximal number of noncrossing straight edges joining them. A polyhedral surface is a threedimensional structure consisting of flat polygonal faces that are joined by common edges. For each of these structures there exist several methods of reconfiguration. Any such method must provide a welldefined way of transforming one instance of a structure to any other. Several types of reconfigurations are reviewed in the introduction, which is followed by new results. We begin with efficient algorithms for comparing monotone chains. Next, we prove that flat chains with unitlength edges and angles within a wide range always admit reconfigurations, under the dihedral model of motion. In this model, angles and edge lengths are preserved. For the universal
Diameter of the thick part of moduli space and simultaneous whitehead moves
 http://arxiv.org/abs/1108.4150
"... Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the thick part of moduli space of S equipped with the Teichmüller or Thurston’s Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order log.gCp /. The same resul ..."
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Cited by 4 (1 self)
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Let S be a surface of genus g with p punctures with negative Euler characteristic. We study the diameter of the thick part of moduli space of S equipped with the Teichmüller or Thurston’s Lipschitz metric. We show that the asymptotic behaviors in both metrics are of order log.gCp /. The same result also holds for the thick part of the moduli space of metric graphs of rank n equipped with the Lipschitz metric. The proof involves a sorting algorithm that sorts an arbitrarily labeled tree with n labels using simultaneous Whitehead moves, where the number of steps is of order log.n/. As a related combinatorial problem, we also compute, in the appendix of this paper, the asymptotic diameter of the moduli space of pants decompositions on S in the metric of elementary moves. 1.
On Local Transformations in Plane Geometric Graphs Embedded on Small Grids (Extended Abstract)
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Reconfiguring triangulations with edge flips and point moves
 In Proceedings of the 12th International Symposium on Graph Drawing, Lecture Notes in Computer Science 3383
, 2004
"... Abstract. We examine reconfigurations between triangulations and neartriangulations of point sets, and give new bounds on the number of point moves and edge flips sufficient for any reconfiguration. We show that with O(n log n) edge flips and point moves, we can transform any geometric neartriangul ..."
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Cited by 3 (3 self)
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Abstract. We examine reconfigurations between triangulations and neartriangulations of point sets, and give new bounds on the number of point moves and edge flips sufficient for any reconfiguration. We show that with O(n log n) edge flips and point moves, we can transform any geometric neartriangulation on n points to any other geometric neartriangulation on n possibly different points. This improves the previously known bound of O(n 2) edge flips and point moves. 1
Induced Subgraphs of Bounded Degree and Bounded Treewidth ⋆
"... Abstract. We prove that for all 0 ≤ t ≤ k and d ≥ 2k, every graph G with treewidth at most k has a ‘large ’ induced subgraph H, where H has treewidth at most t and every vertex in H has degree at most d in G. The order of H depends on t, k, d, and the order of G. With t = k, we obtain large sets of ..."
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Abstract. We prove that for all 0 ≤ t ≤ k and d ≥ 2k, every graph G with treewidth at most k has a ‘large ’ induced subgraph H, where H has treewidth at most t and every vertex in H has degree at most d in G. The order of H depends on t, k, d, and the order of G. With t = k, we obtain large sets of bounded degree vertices. With t = 0, we obtain large independent sets of bounded degree. In both these cases, our bounds on the order of H are tight. For bounded degree independent sets in trees, we characterise the extremal graphs. Finally, we prove that an interval graph with maximum clique size k has a maximum independent set in which every vertex has degree at most 2k. 1
Arc Diagrams, Flip Distances, and Hamiltonian
"... We show that every triangulation (maximal planar graph) on n ≥ 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4connectivity as a means to establish Hamiltonicity. But in general about 3n/5 f ..."
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We show that every triangulation (maximal planar graph) on n ≥ 6 vertices can be flipped into a Hamiltonian triangulation using a sequence of less than n/2 combinatorial edge flips. The previously best upper bound uses 4connectivity as a means to establish Hamiltonicity. But in general about 3n/5 flips are necessary to reach a 4connected triangulation. Our result improves the upper bound on the diameter of the flip graph of combinatorial triangulations on n vertices from 5.2n − 33.6 to 5n − 23. We also show that for every triangulation on n vertices there is a simultaneous flip of less than 2n/3 edges to a 4connected triangulation. The bound on the number of edges is tight, up to an additive constant. As another application we show that every planar graph on n vertices admits an arc diagram with less than n/2 biarcs, that is, after subdividing less than n/2 (of potentially 3n − 6) edges the resulting graph admits a 2page book embedding.
Making triangulations 4connected using flips
, 2011
"... We show that any triangulation on n vertices can be transformed into a 4connected one using at most ⌊(3n − 6)/5 ⌋ edge flips. We also give an example of a triangulation that requires ⌈(3n−10)/5 ⌉ flips to be made 4connected, showing that our bound is tight. Our result implies a new upper bound on ..."
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We show that any triangulation on n vertices can be transformed into a 4connected one using at most ⌊(3n − 6)/5 ⌋ edge flips. We also give an example of a triangulation that requires ⌈(3n−10)/5 ⌉ flips to be made 4connected, showing that our bound is tight. Our result implies a new upper bound on the diameter of the flip graph of 5.2n − 24.4, improving on the bound of 6n − 30 by Mori et al. [4].