Flip graphs of coloured triangulations of convex polygons

Flipping edge-labelled triangulations

Counting thin and bushy triangulations of convex polygons

A polygon with n nodes can be divided into two subpolygons by an internal diagonal through node n. Splitting the polygon along diagonal δi,n and diagonal δn−i,n, i∈{2,…,⌊n/2⌋} results in mirror images. Obviously, there are ⌊n/2⌋−1 pairs of these reflectively symmetrical images. The influence of the observed symmetry on polygon triangulation is studied. The central result of this research is the construction of an efficient algorithm used for generating convex polygon triangulations in minimal time and without generating repeat triangulations. The proposed algorithm uses the diagonal values of the Catalan triangle to avoid duplicate triangulations with negligible computational costs and provides significant speedups compared to known methods.

We prove that the greedy triangulation heuristic for minimum weight triangulation of convex polygons yields solutions within a constant factor from the optimum. For interesting classes of convex polygons, we derive small upper bounds on the constant approximation factor. Our results contrast with Kirkpatrick's Ω(n) bound on the approximation factor of the Delaunay triangulation heuristic for minimum weight triangulation of convexn-vertex polygons. On the other hand, we present a straightforward implementation of the greedy triangulation heuristic for ann-vertex convex point set or a convex polygon takingO(n 2) time andO(n) space. To derive the latter result, we show that given a convex polygonP, one can find for all verticesv ofP a shortest diagonal ofP incident tov in linear time. Finally, we observe that the greedy triangulation for convex polygons having so-called semicircular property can be constructed in timeO(n logn).

This work is devoted to the understanding of properties of random graphs from graph classes with structural constraints. We propose a method that is based on the analysis of the behaviour of Boltzmann sampler algorithms, and may be used to obtain precise estimates for the maximum degree and maximum size of a biconnected block of a "typical'' member of the class in question. We illustrate how our method works on several graph classes, namely dissections and triangulations of convex polygons, embedded trees, and block and cactus graphs.

In this paper, we deal with the problem of generating all triangulations of plane graphs. We give an algorithm for generating all triangulations of a triconnected plane graph G of n vertices. Our algorithm establishes a tree structure among the triangulations of G, called the “tree of triangulations,” and generates each triangulation of G in O(1) time. The algorithm uses O(n) space and generates all triangulations of G without duplications. To the best of our knowledge, our algorithm is the first algorithm for generating all triangulations of a triconnected plane graph; although there exist algorithms for generating triangulated graphs with certain properties. Our algorithm for generating all triangulations of a triconnected plane graph needs to find all triangulations of a convex polygon. We give an algorithm to generate all triangulations of a convex polygon P of n vertices in time O(1) per triangulation, where the vertices of P are numbered. Our algorithm for generating all triangulations of a convex polygon also improves previous results; existing algorithms need to generate all triangulations of convex polygons of less than n vertices before generating the triangulations of a convex polygon of n vertices. Finally, we give an algorithm for generating all triangulations of a convex polygon P of n vertices in time O(n 2) per triangulation, where vertices of P are not numbered.KeywordsPolygonTriangulationGraphPlane GraphGenealogical Tree

Geometric classification of triangulations and their enumeration in a convex polygon

NEW RESULTS IN COMPUTATIONAL GEOMETRY RELEVANT TO PATTERN RECOGNITION IN PRACTICE

LetP andQ be two convex polygons withm andn vertices, respectively, which are specified by their cartesian coordinates in order. A simpleO(m+n) algorithm is presented for computing the intersection ofP andQ. Unlike previous algorithms, the new algorithm consists of a two-step combination of two simple algorithms for finding convex hulls and triangulations of polygons.

Catalan numbers, like Fibonacci and Lucas numbers, appear in a variety of situ ations, including the enumeration of triangulations of convex polygons, well-formed sequences of parentheses, binary trees, and the ballot problem [l]-[5]. Like the other families, Catalan numbers are a great source of pleasure, and are excellent candidates for exploration, experimentation, and conjecturing. They are named after the Belgian mathematician Eugene Catalan (1814-1894), who discovered them in his study of well-formed sequences of parentheses. However, Leon hard Euler (1707-1783) had found them fifty years earlier while counting the number of triangulations of convex polygons [3]. But the credit for the earliest known discov ery goes to the Chinese mathematician Antu Ming (ca. 1692-1763), who was aware of them as early as 1730 [6]. In 1759 the German mathematician and physicist Johann Andreas von Segner (1707-1777), a contemporary of Euler, found that the number Cn of triangulations of a convex polygon satisfies the recursive formula

A linear-time heuristic for minimum weight triangulation of convex polygons is presented. This heuristic produces a triangulation of length within a factor 1 + e from the optimum, where e is an arbitrarily small positive constant. This is the first subcubic algorithm that guarantees such an approximation factor, and it has interesting applications.

Triangulation of the polygon is a fundamental algorithm in computational geometry. This paper considers techniques of object-oriented analysis and design as a new tool for solving and analyzing convex polygon triangulation. The triangulation is analyzed from three aspects: forward, reverse and round-trip engineering. We give a suggestion for improving the obtained software solution of the polygon triangulation algorithm using technique that combines UML modeling and Java programming.

In this paper an algorithm for the convex polygon triangulation based on the reverse Polish notation is proposed. The formal grammar method is used as the starting point in the investigation. This idea is "translated" to the arithmetic expression field enabling application of the reverse Polish notation method. .

This study presents a practical view of dynamic programming, specifically in the context of the application of finding the optimal solutions for the polygon triangulation problem. The problem of the optimal triangulation of polygon is considered to be as a recursive substructure. The basic idea of the constructed method lies in finding to an adequate way for a rapid generation of optimal triangulations and storing - them in as small as possible memory space. The upgraded method is based on a memoization technique, and its emphasis is in storing the results of the calculated values and returning the cached result when the same values again occur. The significance of the method is in the generation of the optimal triangulation for a large number of n. All the calculated weights in the triangulation process are stored and performed in the same table. Results processing and implementation of the method was carried out in the Java environment and the experimental results were compared with the square matrix and Hurtado-Noy method.

Minimal Triangulations of Polygonal Domains