Computational Geometry Column 43

The concept of pointed pseudo-triangulations is defined and a few of its applications described.Comment: 3 pages, 1 figur

On the Number of Pseudo-Triangulations of Certain Point Sets

We pose a monotonicity conjecture on the number of pseudo-triangulations of any planar point set, and check it on two prominent families of point sets, namely the so-called double circle and double chain. The latter has asymptotically $12^n n^{\Theta(1)}$ pointed pseudo-triangulations, which lies significantly above the maximum number of triangulations in a planar point set known so far.Comment: 31 pages, 11 figures, 4 tables. Not much technical changes with respect to v1, except some proofs and statements are slightly more precise and some expositions more clear. This version has been accepted in J. Combin. Th. A. The increase in number of pages from v1 is mostly due to formatting the paper with "elsart.cls" for Elsevie

Expansive Motions and the Polytope of Pointed Pseudo-Triangulations

We introduce the polytope of pointed pseudo-triangulations of a point set in the plane, defined as the polytope of infinitesimal expansive motions of the points subject to certain constraints on the increase of their distances. Its 1-skeleton is the graph whose vertices are the pointed pseudo-triangulations of the point set and whose edges are flips of interior pseudo-triangulation edges. For points in convex position we obtain a new realization of the associahedron, i.e., a geometric representation of the set of triangulations of an n-gon, or of the set of binary trees on n vertices, or of many other combinatorial objects that are counted by the Catalan numbers. By considering the 1-dimensional version of the polytope of constrained expansive motions we obtain a second distinct realization of the associahedron as a perturbation of the positive cell in a Coxeter arrangement. Our methods produce as a by-product a new proof that every simple polygon or polygonal arc in the plane has expansive motions, a key step in the proofs of the Carpenter's Rule Theorem by Connelly, Demaine and Rote (2000) and by Streinu (2000).Comment: 40 pages, 7 figures. Changes from v1: added some comments (specially to the "Further remarks" in Section 5) + changed to final book format. This version is to appear in "Discrete and Computational Geometry -- The Goodman-Pollack Festschrift" (B. Aronov, S. Basu, J. Pach, M. Sharir, eds), series "Algorithms and Combinatorics", Springer Verlag, Berli

Pseudo-Triangulations, Rigidity and Motion Planning

This paper proposes a combinatorial approach to planning non-colliding trajectories for a polygonal bar-and-joint framework with n vertices. It is based on a new class of simple motions induced by expansive one-degree-of-freedom mechanisms, which guarantee noncollisions by moving all points away from each other. Their combinatorial structure is captured by pointed pseudo-triangulations, a class of embedded planar graphs for which we give several equivalent characterizations and exhibit rich rigidity theoretic properties. The main application is an efficient algorithm for the Carpenter\u27s Rule Problem: convexify a simple bar-and-joint planar polygonal linkage using only non-self-intersecting planar motions. A step of the algorithm consists in moving a pseudo-triangulation-based mechanism along its unique trajectory in configuration space until two adjacent edges align. At the alignment event, a local alteration restores the pseudo-triangulation. The motion continues for O(n3) steps until all the points are in convex position. © 2005 Springer Science+Business Media, Inc

Spatial arrangements in architecture and mechanical engineering: some aspects of their representation and construction

Spatial arrangements in architecture and mechanical engineering are represented by incidence structures and classified according to properties of these incidence structures. The relationships between classes are given by ornamentation operations and the construction of elements in fundamental classes by substructure replacement operations. Thus representations of the spatial arrangements for possible designs are generated. Planar maps represent spatial arrangements in architecutral plans. The edges correspond to walls and vertices to incidence between walls. Plans represented by 3-vertex connected maps are ornamented by rooting and extension operations. Further ornamentation specifies access between regions. Plans with all regions adjacent to the exterior correspond to outerplane maps. Trivalent maps represent an important class of plans. Fundamental plans with r internal regions and s regions adjacent to the exterior are represented by [r,s] triangulations. Ornamentations of simple [r,s] triangulations are specified which represent plans with rectangular regions. Plans with walls aligned along two directions are represented by rectangular shapes whose maximal lines correspond to contiguous aligned walls. Rules of construction for various classes are given and the incidence structures of maximal lines and regions are characterized. Spatial arrangements in machines are represented by systems whose blocks correspond to links and vertices to joints. The dual systems are also used. Coplanar kinematic chains with revolute pairs are classified according to mobility and connectedness. Two fundamental classes are considered. First, the chains with binary joints, represented by simple graphs and constructed by two new methods: (i) suspended chain and cycle addition and (ii) subgraph replacement. Second, the chains with binary links which are constructed by subgraph replacement

Selected topics in algorithmic geometry

Let P be a set of n points on the plane with no three points on a line. A crossing-free structure on P is a straight-edge plane graph whose vertex set is P. In this thesis we consider problems of two different topics in the area of algorithmic geometry: Geometry using Steiner points, and counting algorithms. These topics have certain crossing-free structures on P as our primary objects of study. Our results can roughly be described as follows: i) Given a k-coloring of P, with k >= 3 colors, we will show how to construct a set of Steiner points S = S(P) such that a k-colored quadrangulation can always be constructed on (P U S). The bound we show of |S| significantly improves on previously known results. ii) We also show how to construct a se S = S(P) of Steiner points such that a triangulation of (P U S) having all its vertices of even (odd) degree can always be constructed. We show that |S| <= n/3 + c, where c is a constant. We also look at other variants of this problem. iii) With respect to counting algorithms, we show new algorithms for counting triangulations, pseudo-triangulations, crossing-free matchings and crossing-free spanning cycles on P. Our algorithms are simple and allow good analysis of their running times. These algorithms significantly improve over previously known results. We also show an algorithm that counts triangulations approximately, and a hardness result of a particular instance of the problem of counting triangulations exactly. iv) We show experiments comparing our algorithms for counting triangulations with another well-known algorithm that is supposed to be very fast in practice.Sei P eine Menge von n Punkte in der Ebene, so dass keine drei Punkten auf einer Geraden liegen. Eine kreuzungsfreie Struktur von P ist ein geradliniger ebener Graph, der P als Knotenmenge hat. In dieser Dissertation behandeln wir zwei verschiedene Problemkreise auf dem Gebiet der algorithmischen Geometrie: Geometrie mit Steinerpunkten und Anzahl bestimmende Algorithmen auf P und auf gewissen kreuzungsfreien Strukturen von P. Unsere Resultate können wie folgt beschrieben werden: i) Gegeben sei eine k-Färbung von P, mit k >= 3 Farben. Es wird gezeigt, wie eine Menge S = S(P) von Steiner Punkten konstruiert werden kann, die die Konstruktion einer k-gefärbten Quadrangulierung von (P U S) ermöglicht. Die von uns gezeigte Schranke für |S| verbessert die bisher bekannte Schranke. ii) Gezeigt wird auch die Konstruktion einer Menge S = S(P) von Steiner Punkten, so dass eine Triangulierung von (P U S) konstruiert werden kann, bei der der Grad aller Knoten gerade (ungerade) ist. Wir zeigen, dass |S| <= n/3 + c möglich ist, wobei c eine Konstante ist. Wir betrachten auch andere Varianten dieses Problems. iii) Was die Anzahl bestimmenden Algorithmen betrifft, zeigen wir neue Algorithmen, um Triangulierungen, Pseudotriangulierungen, kreuzungsfreie Matchings und kreuzungsfreie aufspannende Zyklen von P zu zählen. Unsere Algorithmen sind einfach und lassen eine gute Analyse der Laufzeiten zu. Diese neuen Algorithmen verbessern wesentlich die bisherigen Ergebnisse. Weiter zeigen wir einen Algorithmus, der Triangulierungen approximativ zählt, und bestimmen die Komplexitätsklasse einer bestimmten Variante des Problems des exakten Zählens von Triangulierungen. iv) Wir zeigen Experimente, die unsere triangulierungszählenden Algorithmen mit einem anderen bekannten Algorithmus vergleichen, der in der Praxis als besonders schnell bekannt ist

Interval Edge-Colorings of Graphs

A proper edge-coloring of a graph G by positive integers is called an interval edge-coloring if the colors assigned to the edges incident to any vertex in G are consecutive (i.e., those colors form an interval of integers). The notion of interval edge-colorings was first introduced by Asratian and Kamalian in 1987, motivated by the problem of finding compact school timetables. In 1992, Hansen described another scenario using interval edge-colorings to schedule parent-teacher conferences so that every person\u27s conferences occur in consecutive slots. A solution exists if and only if the bipartite graph with vertices for parents and teachers, and edges for the required meetings, has an interval edge-coloring. A well-known result of Vizing states that for any simple graph G, χ0(G) ≤ ∆(G)+1, where χ0(G) and ∆(G) denote the edge-chromatic number and maximum degree of G, respectively. A graph G is called class 1 if χ0(G) = ∆(G), and class 2 if χ0(G) = ∆(G) + 1. One can see that any graph admitting an interval edge-coloring must be of class 1, and thus every graph of class 2 does not have such a coloring. Finding an interval edge-coloring of a given graph is hard. In fact, it has been shown that determining whether a bipartite graph has an interval edge-coloring is NP-complete. In this thesis, we survey known results on interval edge-colorings of graphs, with a focus on the progress of (a, b)-biregular bipartite graphs. Discussion of related topics and future work is included at the end. We also give a new proof of Theorem 3.15 on the existence of proper path factors of (3, 4)-biregular graphs. Finally, we obtain a new result, Theorem 3.18, which states that if a proper path factor of any (3, 4)-biregular graph has no path of length 8, then it contains paths of length 6 only. The new result we obtained and the method we developed in the proof of Theorem 3.15 might be helpful in attacking the open problems mentioned in the Future Work section of Chapter 5

Generating spherical multiquadrangulations by restricted vertex splittings and the reducibility of equilibrium classes

A quadrangulation is a graph embedded on the sphere such that each face is bounded by a walk of length 4, parallel edges allowed. All quadrangulations can be generated by a sequence of graph operations called vertex splitting, starting from the path P_2 of length 2. We define the degree D of a splitting S and consider restricted splittings S_{i,j} with i <= D <= j. It is known that S_{2,3} generate all simple quadrangulations. Here we investigate the cases S_{1,2}, S_{1,3}, S_{1,1}, S_{2,2}, S_{3,3}. First we show that the splittings S_{1,2} are exactly the monotone ones in the sense that the resulting graph contains the original as a subgraph. Then we show that they define a set of nontrivial ancestors beyond P_2 and each quadrangulation has a unique ancestor. Our results have a direct geometric interpretation in the context of mechanical equilibria of convex bodies. The topology of the equilibria corresponds to a 2-coloured quadrangulation with independent set sizes s, u. The numbers s, u identify the primary equilibrium class associated with the body by V\'arkonyi and Domokos. We show that both S_{1,1} and S_{2,2} generate all primary classes from a finite set of ancestors which is closely related to their geometric results. If, beyond s and u, the full topology of the quadrangulation is considered, we arrive at the more refined secondary equilibrium classes. As Domokos, L\'angi and Szab\'o showed recently, one can create the geometric counterparts of unrestricted splittings to generate all secondary classes. Our results show that S_{1,2} can only generate a limited range of secondary classes from the same ancestor. The geometric interpretation of the additional ancestors defined by monotone splittings shows that minimal polyhedra play a key role in this process. We also present computational results on the number of secondary classes and multiquadrangulations.Comment: 21 pages, 11 figures and 3 table

Combinatorics, Probability and Computing

One of the exciting phenomena in mathematics in recent years has been the widespread and surprisingly effective use of probabilistic methods in diverse areas. The probabilistic point of view has turned out to b