Recognizing Planar Laman Graphs

Laman graphs are the minimally rigid graphs in the plane. We present two algorithms for recognizing planar Laman graphs. A simple algorithm with running time O(n^(3/2)) and a more complicated algorithm with running time O(n log^3 n) based on involved planar network flow algorithms. Both improve upon the previously fastest algorithm for general graphs by Gabow and Westermann [Algorithmica, 7(5-6):465 - 497, 1992] with running time O(n sqrt{n log n}). To solve this problem we introduce two algorithms (with the running times stated above) that check whether for a directed planar graph G, disjoint sets S, T subseteq V(G), and a fixed k the following connectivity condition holds: for each vertex s in S there are k directed paths from s to T pairwise having only vertex s in common. This variant of connectivity seems interesting on its own

Combinatorial and Geometric Properties of Planar Laman Graphs

Laman graphs naturally arise in structural mechanics and rigidity theory. Specifically, they characterize minimally rigid planar bar-and-joint systems which are frequently needed in robotics, as well as in molecular chemistry and polymer physics. We introduce three new combinatorial structures for planar Laman graphs: angular structures, angle labelings, and edge labelings. The latter two structures are related to Schnyder realizers for maximally planar graphs. We prove that planar Laman graphs are exactly the class of graphs that have an angular structure that is a tree, called angular tree, and that every angular tree has a corresponding angle labeling and edge labeling. Using a combination of these powerful combinatorial structures, we show that every planar Laman graph has an L-contact representation, that is, planar Laman graphs are contact graphs of axis-aligned L-shapes. Moreover, we show that planar Laman graphs and their subgraphs are the only graphs that can be represented this way. We present efficient algorithms that compute, for every planar Laman graph G, an angular tree, angle labeling, edge labeling, and finally an L-contact representation of G. The overall running time is O(n^2), where n is the number of vertices of G, and the L-contact representation is realized on the n x n grid.Comment: 17 pages, 11 figures, SODA 201

Euclidean distance geometry and applications

Euclidean distance geometry is the study of Euclidean geometry based on the concept of distance. This is useful in several applications where the input data consists of an incomplete set of distances, and the output is a set of points in Euclidean space that realizes the given distances. We survey some of the theory of Euclidean distance geometry and some of the most important applications: molecular conformation, localization of sensor networks and statics.Comment: 64 pages, 21 figure

Restricted String Representations

A string representation of a graph assigns to every vertex a curve in the plane so that two curves intersect if and only if the represented vertices are adjacent. This work investigates string representations of graphs with an emphasis on the shapes of curves and the way they intersect. We strengthen some previously known results and show that every planar graph has string representations where every curve consists of axis-parallel line segments with at most two bends (those are the so-called $B_2$-VPG representations) and simultaneously two curves intersect each other at most once (those are the so-called 1-string representations). Thus, planar graphs are $B_2$-VPG $1$-string graphs. We further show that with some restrictions on the shapes of the curves, string representations can be used to produce approximation algorithms for several hard problems. The $B_2$-VPG representations of planar graphs satisfy these restrictions. We attempt to further restrict the number of bends in VPG representations for subclasses of planar graphs, and investigate $B_1$-VPG representations. We propose new classes of string representations for planar graphs that we call ``order-preserving.'' Order-preservation is an interesting property which relates the string representation to the planar embedding of the graph, and we believe that it might prove useful when constructing string representations. Finally, we extend our investigation of string representations to string representations that require some curves to intersect multiple times. We show that there are outer-string graphs that require an exponential number of crossings in their outer-string representations. Our construction also proves that 1-planar graphs, i.e., graphs that are no longer planar, yet fairly close to planar graphs, may have string representations, but they are not always 1-string

Discrete Geometry

The workshop on Discrete Geometry was attended by 53 participants, many of them young researchers. In 13 survey talks an overview of recent developments in Discrete Geometry was given. These talks were supplemented by 16 shorter talks in the afternoon, an open problem session and two special sessions. Mathematics Subject Classification (2000): 52Cxx. Abstract regular polytopes: recent developments. (Peter McMullen) Counting crossing-free configurations in the plane. (Micha Sharir) Geometry in additive combinatorics. (József Solymosi) Rigid components: geometric problems, combinatorial solutions. (Ileana Streinu) • Forbidden patterns. (János Pach) • Projected polytopes, Gale diagrams, and polyhedral surfaces. (Günter M. Ziegler) • What is known about unit cubes? (Chuanming Zong) There were 16 shorter talks in the afternoon, an open problem session chaired by Jesús De Loera, and two special sessions: on geometric transversal theory (organized by Eli Goodman) and on a new release of the geometric software Cinderella (Jürgen Richter-Gebert). On the one hand, the contributions witnessed the progress the field provided in recent years, on the other hand, they also showed how many basic (and seemingly simple) questions are still far from being resolved. The program left enough time to use the stimulating atmosphere of the Oberwolfach facilities for fruitful interaction between the participants

Direct tree decomposition of geometric constraint graphs

The evolution of constraint based geometric models is tightly tied to parametric and feature-based Computer-Aided Design (CAD) systems. Since the introduction of parametric design by Pro/Engineer in the 1980's, most major CAD systems adopted constraint based geometric models as a core technology. Constraint based geometric models allowed CAD systems to provide a more powerful data model while offering an intuitive user interface. Later on, the same models also found application to fields like linkage design, chemical modeling, computer vision and dynamic geometry. Constraint based geometric models are unevaluated models. A key problem related to constraint based geometric models is the geometric constraint based solving problem which, roughly speaking, can be stated as the problem of evaluating a constraint based model. Among the different approaches to geometric constraint solving, we are interested in graph-based Decomposition-Recombination solvers. In the graph-based constructive approach, the geometric problem is first translated into a graph whose vertices represent the set of geometric elements and whose edges are the constraints. Then the constraint problem is solved by decomposing the graph into a set of sub-problems, each sub-problem is recursively divided until reaching basic problems which are solved by a dedicated equational solver. The solution to the initial problem is computed by merging the solutions to the sub-problems. The approach used by DR-solvers has been particularly successful when the decomposition into subproblems and subsequent recombination of solutions to these subproblems can be described by a plan generated a priori, that is, a plan generated as a preprocessing step without actually solving the subsystems. The plan output by the DR-planner remains unchanged as numerical values of parameters change. Such a plan is known as a DR-plan and the unit in the solver that generates it is the DR-planner. In this setting, the DR-plan is then used to drive the actual solving process, that is, computing specific coordinates that properly place geometric objects with respect to each other. In this thesis we develop a new DR-planner algorithm for graph-constructive two dimensional DR-solvers. This DR-planner is based on the tree-decomposition of a graph. The triangle- or tree-decomposition of a graph decomposes a graph into three subgraphs such that subgraphs pairwise share one vertex. Shared vertices are called hinges. The tree-decomposition of a geometric constraint graph is in some sense the construction plan that solves the corresponding problem. The DR-planner algorithm first transforms the input graph into a simpler, planar graph. After that, an specific planar embedding is computed for the transformed graph where hinges, if any, can be straightly found. In the work we proof the soundness of the new algorithm. We also show that the worst case time performance of the resthe number of vertices of the input graph. The resulting algorithm is easy to implement and is as efficient as other known solving algorithms.L'evolució de models geomètrics basats en restriccions està fortament lligada al sistemes de Disseny Assistit per Computador (CAD) paramètrics i als basats en el paradigma de disseny per mitjà de característiques. Des de la introducció del disseny paramètric per part de Pro/Engineer en els anys 80, la major part de sistemes CAD utilitzaren com a tecnologia de base els models geomètrics basats en restriccions. Els models geomètrics basats en restriccions permeteren als sistemes CAD proporcionar un model d'informació més ampli i alhora oferir una interfície d'usuari intuitiva. Posteriorment, els mateixos models s'aplicaren en camps com el disseny de mecanismes, el modelatge químic, la visió per computador i la geometria dinàmica. Els models geomètrics basats en restriccions són models no avaluats. Un problema clau relacionat amb el models de restriccions geomètriques és el problema de la resolució de restriccions geomètriques, que es resumeix com el problema d'avaluar un model basat en restriccions. Entre els diferents enfocs de resolució de restriccions geomètriques, tractem els solvers de Descomposició-Recombinació (DR-solvers) basats en graphs. En l'enfoc constructiu basat en grafs, el problema geomètric es trasllada en un pas inicial a un graf, on els vèrtexs del graf representen el conjunt d'elements geomètrics i on les arestes corresponen a les restriccions geomètriques entre els elements. A continuació el problema de restriccions es resol descomposant el graf en un conjunt de subproblemes, cadascun dels quals es divideix recursivament fins a obtenir problemes bàsics, que sovint són operacions geomètriques realitzables, per exemple, amb regle i compàs, i que es resolen per mitjà d'un solver numèric específic. Finalment, la solució del problema inicial s'obté recombinant les solucions dels subproblemes. L'enfoc utilitzat pels DR-solvers ha esdevingut especialment interessant quan la descomposició en subproblemes i la posterior recombinació de solucions d'aquests subproblemes es pot descriure com un pla de construcció generat a priori, és a dir, un pla generat com a pas de pre-procés sense necessitat de resoldre realment els subsistemes. El pla generat pel DR-planner esdevé inalterable encara que els valors numèrics dels paràmetres canviin. Aquest pla es coneix com a DR-plan i la unitat en el solver que el genera és l'anomenat DR-planner. En aquest context, el DR-plan s'utilitza com a eina del procés de resolució en curs, és a dir, permet calcular les coordenades específiques que correctament posicionen els elements geomètrics uns respecte els altres. En aquesta tesi desenvolupem un nou algoritme que és la base del DR-planner per a DR-solvers constructius basats en grafs en l'espai bidimensional. Aquest DR-planner es basa en la descomposició en arbre d'un graf. La descomposició en triangles o arbre de descomposició d'un graf es basa en descomposar un graf en tres subgrafs tals que comparteixen un vèrtex 2 a 2. El conjunt de vèrtexs compartits s'anomenen \emph{hinges}. La descomposició en arbre d'un graf de restriccions geomètriques equival, en cert sentit, a resoldre el problema de restriccions geomètriques. L'algoritme del DR-planner en primer lloc transforma el graf proporcionat en un graf més simple i planar. A continuació, es calcula el dibuix en el pla del graf transformat, on les hinges, si n'hi ha, es calculen de manera directa. En aquest treball demostrem la correctesa del nou algoritme. Finalment, proporcionem l'estudi de la complexitat temporal de l'algoritme en cas pitjor i demostrem que és quadràtica en el nombre de vèrtexs del graf proporcionat. L'algoritme resultant és senzill d'implementar i tan eficient com altres algoritmes de resolució concret