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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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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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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
On Local Transformations in Plane Geometric Graphs Embedded on Small Grids (Extended Abstract)
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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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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.
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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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
Making triangulations 4connected using flipsI
"... We show that any combinatorial triangulation on n vertices can be transformed into a 4connected one using at most b(3n−9)/5c edge flips. We also give an example of an infinite family of triangulations that requires this many flips to be made 4connected, showing that our bound is tight. In additio ..."
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We show that any combinatorial triangulation on n vertices can be transformed into a 4connected one using at most b(3n−9)/5c edge flips. We also give an example of an infinite family of triangulations that requires this many flips to be made 4connected, showing that our bound is tight. In addition, for n ≥ 19, we improve the upper bound on the number of flips required to transform any 4connected triangulation into the canonical triangulation (the triangulation with two dominant vertices), matching the known lower bound of 2n − 15. Our results imply a new upper bound on the diameter of the flip graph of 5.2n − 33.6, improving on the previous best known bound of 6n − 30.
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].
October 8, 2007 14:22 WSPC/Guidelines EdgeContractionOct08 Restricted Mesh Simplification Using Edge Contractions
"... We consider the problem of simplifying a planar triangle mesh using edge contractions, under the restriction that the resulting vertices must be a subset of the input set. That is, contraction of an edge must be made onto one of its adjacent vertices, which results in removing the other adjacent ver ..."
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We consider the problem of simplifying a planar triangle mesh using edge contractions, under the restriction that the resulting vertices must be a subset of the input set. That is, contraction of an edge must be made onto one of its adjacent vertices, which results in removing the other adjacent vertex. We show that if the perimeter of the mesh consists of at most five vertices, then we can always find a vertex not on the perimeter which can be removed in this way. If the perimeter consists of more than five vertices such a vertex may not exist. In order to maintain a higher number of removable vertices under the above restriction, we study edge flips which can be performed in a visually smooth way. A removal of a vertex which is preceded by one such smooth operation is called a 2step removal. Moreover, we introduce the possibility that the user defines “important” vertices (or edges) which have to remain intact. Given m such important vertices, or edges, we show that a simplification hierarchy of size O(n) and depth O(log(n/m)) can be constructed by 2step removals in O(n) time, such that the simplified graph contains the m important vertices and edges, and at most O(m) other vertices from the input graph. In some triangulations, many vertices may not even be 2step removable. In order to provide the option to remove such vertices, we also define and examine kstep removals. This increases the lower bound on the number of removable vertices. Keywords: Computational Geometry, Computer graphics, Edge Contractions. 1.