Ferran Hurtado

Flipping Edges in Triangulations

Abstract. In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least n

Flips in planar graphs

lceil (n-4)/2 rceil

Compatible geometric matchings

edges that can be flipped. We also prove that O(n + k2) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T such that to transform T into T requires Ω(n2) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips.

On the number of plane graphs

We review results concerning edge flips in planar graphs concentrating mainly on various aspects of the following problem: Given two different planar graphs of the same size, how many edge flips are necessary and sufficient to transform one graph into another? We overview both the combinatorial perspective (where only a combinatorial embedding of the graph is specified) and the geometric perspective (where the graph is embedded in the plane, vertices are points and edges are straight-line segments). We highlight the similarities and differences of the two settings, describe many extensions and generalizations, highlight algorithmic issues, outline several applications and mention open problems.

Plane Geometric Graph Augmentation: A Generic Perspective

This paper studies non-crossing geometric perfect matchings. Two such perfect matchings are compatible if they have the same vertex set and their union is also non-crossing. Our first result states that for any two perfect matchings M and M^ of the same set of n points, for some k@?O(logn), there is a sequence of perfect matchings M=M0,M1,...,Mk=M^, such that each Mi is compatible with Mi+1. This improves the previous best bound of k=

Graphs of non-crossing perfect matchings

We investigate the number of plane geometric, i.e., straight-line, graphs, a set <i>S</i> of <i>n</i> points in the plane admits. We show that the number of plane graphs and connected plane graphs as well as the number of cycle-free plane graphs is minimized when <i>S</i> is in convex position. Moreover, these results hold for all these graphs with an arbitrary but fixed number of edges. Consequently, we provide simple proofs that the number of spanning trees, cycle-free graphs (forests), perfect matchings, and spanning paths is also minimized for point sets in convex position.In addition we construct a new extremal configuration, the so-called double zig-zag chain. Most noteworthy this example bears Θ*(√72^<i>n</i>) = Θ*(8.4853^<i>n</i>) triangulations and Θ*(41.1889^<i>n</i>) plane graphs (omitting polynomial factors in both cases), improving the previously known best maximizing examples.

Graphs of Triangulations and Perfect Matchings

Graph augmentation problems are motivated by network design and have been studied extensively in optimization. We consider augmentation problems over plane geometric graphs, that is, graphs given with a crossing-free straight-line embedding in the plane. The geometric constraints on the possible new edges render some of the simplest augmentation problems intractable, and in many cases only extremal results are known. We survey recent results, highlight common trends, and gather numerous conjectures and open problems.

Flipping edges in triangulations

Abstract.u2002Let Pn be a set of n=2m points that are the vertices of a convex polygon, and let ℳm be the graph having as vertices all the perfect matchings in the point set Pn whose edges are straight line segments and do not cross, and edges joining two perfect matchings M1 and M2 if M2=M1−(a,b)−(c,d)+(a,d)+(b,c) for some points a,b,c,d of Pn. We prove the following results about ℳm: its diameter is m−1; it is bipartite for every m; the connectivity is equal to m−1; it has no Hamilton path for m odd, m>3; and finally it has a Hamilton cycle for every m even, m≥4.

Matching Points with Squares

Given a set P of points in general position in the plane, the graph of triangulations of P has a vertex for every triangulation of P, and two of them are adjacent if they differ by a single edge exchange. We prove that the subgraph of , consisting of all triangulations of P that admit a perfect matching, is connected. A main tool in our proof is a result of independent interest, namely that the graph that has as vertices the non-crossing perfect matchings of P and two of them are adjacent if their symmetric difference is a single non-crossing cycle, is also connected.

Matching points with circles and squares

In this paper we study the problem of flipping edges in triangulations of polygons and point sets. One of the main results is that any triangulation of a set of n points in general position contains at least (lceil (n-4)/2 rceil) edges that can be flipped. We also prove that O(n + k2) flips are sufficient to transform any triangulation of an n -gon with k reflex vertices into any other triangulation. We produce examples of n -gons with triangulations T and T such that to transform T into T requires Ω(n2) flips. Finally we show that if a set of n points has k convex layers, then any triangulation of the point set can be transformed into any other triangulation using at most O(kn) flips.