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25
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 37 (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 ...
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 36 (6 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.
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 23 (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 7 (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.
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 7 (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...
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 7 (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...
Polytope Skeletons And Paths
 Handbook of Discrete and Computational Geometry (Second Edition ), chapter 20
"... INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent i ..."
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Cited by 6 (0 self)
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INTRODUCTION The kdimensional skeleton of a dpolytope P is the set of all faces of the polytope of dimension at most k. The 1skeleton of P is called the graph of P and denoted by G(P ). G(P ) can be regarded as an abstract graph whose vertices are the vertices of P , with two vertices adjacent if they form the endpoints of an edge of P . In this chapter, we will describe results and problems concerning graphs and skeletons of polytopes. In Section 17.1 we briefly describe the situation for 3polytopes. In Section 17.2 we consider general properties of polytopal graphs subgraphs and induced subgraphs, connectivity and separation, expansion, and other properties. In Section 17.3 we discuss problems related to diameters of polytopal graphs in connection with the simplex algorithm and t
On the pathwidth of planar graphs
 Research report, INRIA Research Report HAL00082035
, 2006
"... In this paper, we present a result concerning the relation between the pathwith of a planar graph and the pathwidth of its dual. More precisely, we prove that for a 3connected planar graph G, pw(G) ≤ 3pw(G ∗ ) + 2. For 4connected planar graphs, and more generally for Hamiltonian planar graphs, ..."
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Cited by 5 (1 self)
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In this paper, we present a result concerning the relation between the pathwith of a planar graph and the pathwidth of its dual. More precisely, we prove that for a 3connected planar graph G, pw(G) ≤ 3pw(G ∗ ) + 2. For 4connected planar graphs, and more generally for Hamiltonian planar graphs, we prove a stronger bound pw(G ∗ ) ≤ 2 pw(G) + c. The best previously known bound was obtained by Fomin and Thilikos who proved that pw(G ∗ ) ≤ 6 pw(G) + cte. The proof is based on an algorithm which, given a fixed spanning tree of G, transforms any given decomposition of G into one of G ∗. The ratio of the corresponding parameters is bounded by the maximum degree of the spanning tree. 1
Long Cycles in Graphs on a Fixed Surface
, 1999
"... We prove that there exists a function a : N 0 \Theta R+ ! N such that ..."
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Cited by 3 (1 self)
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We prove that there exists a function a : N 0 \Theta R+ ! N such that