A straight-line drawing δ of a planar graph G need not be plane but can be made so by untangling it, that is, by moving some of the vertices of G. Let shift(G,δ) denote the minimum number of vertices that need to be moved to untangle δ. We show that shift(G,δ) is NP-hard to compute and to approximate. Our hardness results extend to a version of 1BendPointSetEmbeddability, a ...

An Apollonian configuration of circles is a collection of circles in the plane with disjoint interiors such that the complement of the interiors of the circles consists of curvilinear triangles. One well-studied method of forming an Apollonian configuration is to start with three mutually tangent circles and fill a curvilinear triangle with a new circle, then repeat with each newly ...

We show that the vertices of any plane graph in which every face is incident to at least g vertices can be colored by ⌊(3g−5)/4⌋ colors so that every color appears in every face. This is nearly tight, as there are plane graphs where all faces are incident to at least g vertices and that admit no vertex coloring of this type with more than ⌊(3g+1)/4⌋ colors. We further show that the ...

This paper describes an algorithm for generating a guaranteed intersection-free interpolation sequence between any pair of compatible polygons. Our algorithm builds on prior results from linkage unfolding, and if desired it can ensure that every edge length changes monotonically over the course of the interpolation sequence. The computational machinery that ensures against ...

The purpose of this paper is to establish an inequality connecting the lattice point enumerator of a 0-symmetric convex body with its successive minima. To this end, we introduce an optimization problem whose solution refines former methods, thus producing a better upper bound. In particular, we show that an analogue of Minkowski’s second theorem on successive minima with the ...

We show that the fundamental group of ordered affine-equivalent configurations with at least five points in the real plane is isomorphic to the pure braid group in as many strands, modulo its centre. In the case of four points, this fundamental group is free with 11 generators.

Given a metric M=(V,d), a graph G=(V,E) is a t-spanner for M if every pair of nodes in V has a “short” path (i.e., of length at most t times their actual distance) between them in the spanner. Furthermore, this spanner has a hop diameter bounded by D if every pair of nodes has such a short path that also uses at most D edges. We consider the problem of constructing sparse ...

We introduce a new class of fat, not necessarily convex or polygonal, objects in the plane, namely locally γ-fat objects. We prove that the union complexity of any set of n such objects is O(λ s+2(n)log 2 n). This improves the best known bound, and extends it to a more general class of objects.

We show that generating all negative cycles of a weighted graph is a hard enumeration problem, in both the directed and undirected cases. More precisely, given a family of negative (directed) cycles, it is an NP-complete problem to decide whether this family can be extended or there are no other negative (directed) cycles in the graph, implying that (directed) negative cycles ...

We investigate algorithmic questions that arise in the statistical problem of computing lines or hyperplanes of maximum regression depth among a set of n points. We work primarily with a dual representation and find points of maximum undirected depth in an arrangement of lines or hyperplanes. An O(n d ) time and O(n d−1) space algorithm computes undirected depth of all points in d ...