Happy endings for flip graphs

We show that the triangulations of a finite point set form a flip graph that can be embedded isometrically into a hypercube, if and only if the point set has no empty convex pentagon. Point sets of this type include convex subsets of lattices, points on two lines, and several other infinite families. As a consequence, flip distance in such point sets can be computed efficiently.Comment: 26 pages, 15 figures. Revised and expanded for journal publicatio

On Monotone Sequences of Directed Flips, Triangulations of Polyhedra, and Structural Properties of a Directed Flip Graph

This paper studied the geometric and combinatorial aspects of the classical Lawson's flip algorithm in 1972. Let A be a finite set of points in R2, omega be a height function which lifts the vertices of A into R3. Every flip in triangulations of A can be associated with a direction. We first established a relatively obvious relation between monotone sequences of directed flips between triangulations of A and triangulations of the lifted point set of A in R3. We then studied the structural properties of a directed flip graph (a poset) on the set of all triangulations of A. We proved several general properties of this poset which clearly explain when Lawson's algorithm works and why it may fail in general. We further characterised the triangulations which cause failure of Lawson's algorithm, and showed that they must contain redundant interior vertices which are not removable by directed flips. A special case if this result in 3d has been shown by B.Joe in 1989. As an application, we described a simple algorithm to triangulate a special class of 3d non-convex polyhedra. We proved sufficient conditions for the termination of this algorithm and show that it runs in O(n3) time.Comment: 40 pages, 35 figure

Kinetic Voronoi Diagrams and Delaunay Triangulations under Polygonal Distance Functions

Let $P$ be a set of $n$ points and $Q$ a convex $k$-gon in ${\mathbb R}^2$. We analyze in detail the topological (or discrete) changes in the structure of the Voronoi diagram and the Delaunay triangulation of $P$, under the convex distance function defined by $Q$, as the points of $P$ move along prespecified continuous trajectories. Assuming that each point of $P$ moves along an algebraic trajectory of bounded degree, we establish an upper bound of $O(k^4n\lambda_r(n))$ on the number of topological changes experienced by the diagrams throughout the motion; here $\lambda_r(n)$ is the maximum length of an $(n,r)$-Davenport-Schinzel sequence, and $r$ is a constant depending on the algebraic degree of the motion of the points. Finally, we describe an algorithm for efficiently maintaining the above structures, using the kinetic data structure (KDS) framework

Improvement of the robustness on geographical networks by adding shortcuts

In a topological structure affected by geographical constraints on liking, the connectivity is weakened by constructing local stubs with small cycles, a something of randomness to bridge them is crucial for the robust network design. In this paper, we numerically investigate the effects of adding shortcuts on the robustness in geographical scale-free network models under a similar degree distribution to the original one. We show that a small fraction of shortcuts is highly contribute to improve the tolerance of connectivity especially for the intentional attacks on hubs. The improvement is equivalent to the effect by fully rewirings without geographical constraints on linking. Even in the realistic Internet topologies, these effects are virtually examined.Comment: 14 pages, 10 figures, 1 tabl

Flipping Cubical Meshes

We define and examine flip operations for quadrilateral and hexahedral meshes, similar to the flipping transformations previously used in triangular and tetrahedral mesh generation.Comment: 20 pages, 24 figures. Expanded journal version of paper from 10th International Meshing Roundtable. This version removes some unwanted paragraph breaks from the previous version; the text is unchange