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35
Constructing Plane Spanners of Bounded Degree and Low Weight
 in Proceedings of European Symposium of Algorithms
, 2002
"... Given a set S of n points in the plane, we give an O(n log n)time algorithm that constructs a plane tspanner for S, for t 10:02, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. These c ..."
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Cited by 48 (7 self)
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Given a set S of n points in the plane, we give an O(n log n)time algorithm that constructs a plane tspanner for S, for t 10:02, such that the degree of each point of S is bounded from above by 27, and the total edge length is proportional to the weight of a minimum spanning tree of S. These constants are all worst case constants that are artifacts of our proofs. In practice, we believe them to be much smaller. Previously, no algorithms were known for constructing plane tspanners of bounded degree.
Convex Grid Drawings of 3Connected Planar Graphs
, 1994
"... We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a lineartime algorithm which, given an nvertex 3connected plane gr ..."
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Cited by 42 (7 self)
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We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a lineartime algorithm which, given an nvertex 3connected plane graph G (with n 3), finds such a straightline convex embedding of G into a (n \Gamma 2) \Theta (n \Gamma 2) grid. 1 Introduction In this paper we consider the problem of aesthetic drawing of plane graphs, that is, planar graphs that are already embedded in the plane. What is exactly an aesthetic drawing is not precisely defined and, depending on the application, different criteria have been used. In this paper we concentrate on the two following criteria: (a) edges should be represented by straightline segments, and (b) faces should be drawn as convex polygons. F'ary [6], Stein [14] and Wagner [18] showed, independently, that each planar graph can be drawn in the plane in such a way that ...
Some results on greedy embeddings in metric spaces
 In Proc. of the 49th IEEE Annual Symposium on Foundations of Computer Science
, 2008
"... Geographic Routing is a family of routing algorithms that uses geographic point locations as addresses for the purposes of routing. Such routing algorithms have proven to be both simple to implement and heuristically effective when applied to wireless sensor networks. Greedy Routing is a natural abs ..."
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Cited by 35 (0 self)
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Geographic Routing is a family of routing algorithms that uses geographic point locations as addresses for the purposes of routing. Such routing algorithms have proven to be both simple to implement and heuristically effective when applied to wireless sensor networks. Greedy Routing is a natural abstraction of this model in which nodes are assigned virtual coordinates in a metric space, and these coordinates are used to perform pointtopoint routing. Here we resolve a conjecture of Papadimitriou and Ratajczak that every 3connected planar graph admits a greedy embedding into the Euclidean plane. This immediately implies that all 3connected graphs that exclude K3,3 as a minor admit a greedy embedding into the Euclidean plane. Additionally, we provide the first nontrivial examples of graphs that admit no such embedding. These structural results provide efficiently verifiable certificates that a graph admits a greedy embedding or that a graph admits no greedy embedding into the Euclidean plane.
Long Cycles in 3Connected Graphs
"... Moon and Moser in 1963 conjectured that if G is a 3connected planar graph on n vertices, then G contains a cycle of length at least \Omega (nlog3 2). In this paper, this conjecture is proved. In addition, the same result is proved for 3connected graphs embeddable in the projective plane, or the t ..."
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Cited by 12 (3 self)
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Moon and Moser in 1963 conjectured that if G is a 3connected planar graph on n vertices, then G contains a cycle of length at least \Omega (nlog3 2). In this paper, this conjecture is proved. In addition, the same result is proved for 3connected graphs embeddable in the projective plane, or the torus, or the Klein bottle.
Nets of Polyhedra
, 1997
"... In 1525, the painter Albrecht Dürer introduced the notion of a net of a polytope, and published nets for some of the Platonian and Archimedian polyhedra, along with directions about how to construct them. An unfolding of a 3dimensional polytope P is obtained by cutting the boundary of P along a co ..."
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Cited by 9 (0 self)
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In 1525, the painter Albrecht Dürer introduced the notion of a net of a polytope, and published nets for some of the Platonian and Archimedian polyhedra, along with directions about how to construct them. An unfolding of a 3dimensional polytope P is obtained by cutting the boundary of P along a collection of edges that spans the vertex set of P and then flattening the remaining set to a polygon in the plane. An unfolding is a net if it does not overlap itself. Conversely, a simple connected plane polygon with specific folding lines is a net, if it is possible to fold it into (the boundary of) a polytope. We consider the question whether every 3dimensional polytope has a net. Although the problem is intuitive and easy to state, and there are nets known for all regular and uniform polytopes, in general it is still unsolved. After giving an overview of related questions and conjectures about the nature or existence of nets for 3polytopes, we present an account of our experiments wit...
On 3Connected Plane Graphs Without Triangular Faces
, 1998
"... . We prove that each polyhedral triangular face free map G on a compact 2dimensional manifold M with Euler characteristic Ø(M) contains a kpath, i.e. a path on k vertices, such that each vertex of this path has, in G, degree at most 5 2 k if M is a sphere S 0 and at most k 2 ¯ 5+ p 49\Gam ..."
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Cited by 8 (1 self)
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. We prove that each polyhedral triangular face free map G on a compact 2dimensional manifold M with Euler characteristic Ø(M) contains a kpath, i.e. a path on k vertices, such that each vertex of this path has, in G, degree at most 5 2 k if M is a sphere S 0 and at most k 2 ¯ 5+ p 49\Gamma24Ø(M 2 if M 6= S 0 or does not contain any kpath. We show that for even k this bound is best possible. Moreover, we verify that for any graph other than a path no similar estimation exists. 1. Introduction Throughout this paper we shall consider connected graphs without loops or multiple edges. Let P r denote a path on r vertices (an rpath in the sequel). For graphs H and G, G ¸ = H denotes that the graphs H and G are isomorphic. The standard notation \Delta(G) stands for the maximum degree of a graph G. For a vertex X of a graph G deg G (X) denotes the degree of X in G. Let H be a family of graphs and let H be a graph which is isomorphic to a subgraph of at least one member of H...
Polytope Skeletons And Paths
 HANDBOOK OF DISCRETE AND COMPUTATIONAL GEOMETRY (SECOND EDITION ), CHAPTER 20
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On the Pathwidth of Planar Graphs
, 2006
"... Fomin and Thilikos in [5] conjectured that there is a constant c such that, for every 2connected planar graph G, pw(G) 2pw(G)+c (the same question was asked simutaneously by Coudert, Huc and Sereni in [4]). By the results of Boedlander and Fomin [2] this holds for every outerplanar graph and ac ..."
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Cited by 6 (1 self)
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Fomin and Thilikos in [5] conjectured that there is a constant c such that, for every 2connected planar graph G, pw(G) 2pw(G)+c (the same question was asked simutaneously by Coudert, Huc and Sereni in [4]). By the results of Boedlander and Fomin [2] this holds for every outerplanar graph and actually is tight by Coudert, Huc and Sereni [4]. In [5], Fomin and Thilikos proved that there is a constant c such that the pathwidth of every 3connected graph G satisfies: pw(G) 6pw(G) + c. In this paper we improve this result by showing that the dual a 3connected planar graph has pathwidth at most 3 times the pathwidth of the primal plus two. We prove also that the question can be answered positively for 4connected planar graphs.
Untangling polygons and graphs
, 2008
"... Untangling is a process in which some vertices of a plane graph are moved to obtain a straightline plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph Cn while keeping at least Ω(n 2/3) vertices fixed. For any graph G, we also presen ..."
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Cited by 4 (0 self)
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Untangling is a process in which some vertices of a plane graph are moved to obtain a straightline plane drawing. The aim is to move as few vertices as possible. We present an algorithm that untangles the cycle graph Cn while keeping at least Ω(n 2/3) vertices fixed. For any graph G, we also present an upper bound for the number of fixed vertices in the worst case. The bound is a function of the number of vertices, maximum degree and diameter of G. One of its consequences is the upper bound O((n log n) 2/3) for all 3vertexconnected planar graphs.