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Local Polyhedra and Geometric Graphs*
"... Local Polyhedra and Geometric Graphs 1 1 Introduction Nonconvex polyhedra are ubiquitous in computer graphics, solid modeling, computer aided design and manufacturing, robotics, and other geometric application areas. Unlike nonconvex polygons or convex objects in space, for which many problems can b ..."
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Local Polyhedra and Geometric Graphs 1 1 Introduction Nonconvex polyhedra are ubiquitous in computer graphics, solid modeling, computer aided design and manufacturing, robotics, and other geometric application areas. Unlike nonconvex polygons or convex objects in space, for which many problems can
ALGORITHMS AND COMBINATORICS FOR GEOMETRIC GRAPHS
, 2014
"... These are the course notes for half of the MPRI course “Algorithms and combinatorics for geometric graphs”. Announcements for this course may be found on the webpage ..."
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These are the course notes for half of the MPRI course “Algorithms and combinatorics for geometric graphs”. Announcements for this course may be found on the webpage
Geometric Intersection Patterns and the Theory of Topological Graphs
"... The intersection graph of a set system S is a graph on the vertex set S, in which two vertices are connected by an edge if and only if the corresponding sets have nonempty intersection. It was shown by Tietze (1905) that every finite graph is the intersection graph of 3dimensional convex polytope ..."
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is a graph drawn in the plane so that its vertices are represented by points and its edges by possibly intersecting simple continuous curves connecting the corresponding point pairs. We give applications of the results concerning intersection patterns in the theory of topological graphs.
Generating Geometric Graphs Using Automorphisms
, 2012
"... We outline an algorithm for generating nets, a “net” being a geometric graph embedded in a space on which some group acts transitively. We prove that any connected net in such a space can be generated by this process, which can also be regarded as a covering of the net by copies of a connected “tran ..."
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We outline an algorithm for generating nets, a “net” being a geometric graph embedded in a space on which some group acts transitively. We prove that any connected net in such a space can be generated by this process, which can also be regarded as a covering of the net by copies of a connected
The complexity of simultaneous geometric graph embedding
, 2015
"... Given a collection of planar graphs G1,..., Gk on the same set V of n vertices, the simultaneous geometric embedding (with mapping) problem, or simply kSGE, is to find a set P of n points in the plane and a bijection ϕ: V → P such that the induced straightline drawings of G1,..., Gk under ϕ are al ..."
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Given a collection of planar graphs G1,..., Gk on the same set V of n vertices, the simultaneous geometric embedding (with mapping) problem, or simply kSGE, is to find a set P of n points in the plane and a bijection ϕ: V → P such that the induced straightline drawings of G1,..., Gk under ϕ
Lectures on Geometric Group Theory
"... This book is based upon a set of lecture notes for a course that I was teaching at the University of Utah in Fall of 2002. Our main goal is to describe various tools of the quasiisometric rigidity and to illustrate them by presenting (essentially selfcontained) proofs of several fundamental theore ..."
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theory is to treat finitelygenerated groups as geometric objects: with each finitelygenerated group G we associate a metric space, the Cayley graph of G. One of the main issues of the geometric group theory is to recover as much as possible algebraic information about G from the geometry of the Cayley
ABSTRACT Local Polyhedra and Geometric Graphs ∗
"... We introduce a new realistic input model for geometric graphs and nonconvex polyhedra. A geometric graph G is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor and (2) the lengths of the longest and shortest ed ..."
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We introduce a new realistic input model for geometric graphs and nonconvex polyhedra. A geometric graph G is local if (1) the longest edge at every vertex v is only a constant factor longer than the distance from v to its Euclidean nearest neighbor and (2) the lengths of the longest and shortest
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