There are Plane Spanners of Maximum Degree 4

Let E be the complete Euclidean graph on a set of points embedded in the plane. Given a constant t >= 1, a spanning subgraph G of E is said to be a t-spanner, or simply a spanner, if for any pair of vertices u,v in E the distance between u and v in G is at most t times their distance in E. A spanner is plane if its edges do not cross. This paper considers the question: "What is the smallest maximum degree that can always be achieved for a plane spanner of E?" Without the planarity constraint, it is known that the answer is 3 which is thus the best known lower bound on the degree of any plane spanner. With the planarity requirement, the best known upper bound on the maximum degree is 6, the last in a long sequence of results improving the upper bound. In this paper we show that the complete Euclidean graph always contains a plane spanner of maximum degree at most 4 and make a big step toward closing the question. Our construction leads to an efficient algorithm for obtaining the spanner from Chew's L1-Delaunay triangulation

The Tight Spanning Ratio of the Rectangle Delaunay Triangulation

Spanner construction is a well-studied problem and Delaunay triangulations are among the most popular spanners. Tight bounds are known if the Delaunay triangulation is constructed using an equilateral triangle, a square, or a regular hexagon. However, all other shapes have remained elusive. In this paper we extend the restricted class of spanners for which tight bounds are known. We prove that Delaunay triangulations constructed using rectangles with aspect ratio \A have spanning ratio at most \sqrt{2} \sqrt{1+\A^2 + \A \sqrt{\A^2 + 1}}, which matches the known lower bound

Constrained generalized Delaunay graphs are plane spanners

We look at generalized Delaunay graphs in the constrained setting by introducing line segments which the edges of the graph are not allowed to cross. Given an arbitrary convex shape C, a constrained Delaunay graph is constructed by adding an edge between two vertices p and q if and only if there exists a homothet of C with p and q on its boundary that does not contain any other vertices visible to p and q. We show that, regardless of the convex shape C used to construct the constrained Delaunay graph, there exists a constant t (that depends on C) such that it is a plane t-spanner of the visibility graph

Generalized Delaunay triangulations : graph-theoretic properties and algorithms

This thesis studies different generalizations of Delaunay triangulations, both from a combinatorial and algorithmic point of view. The Delaunay triangulation of a point set S, denoted DT(S), has vertex set S. An edge uv is in DT(S) if it satisfies the empty circle property: there exists a circle with u and v on its boundary that does not enclose points of S. Due to different optimization criteria, many generalizations of the DT(S) have been proposed. Several properties are known for DT(S), yet, few are known for its generalizations. The main question we explore is: to what extent can properties of DT(S) be extended for generalized Delaunay graphs? First, we explore the connectivity of the flip graph of higher order Delaunay triangulations of a point set S in the plane. The order-k flip graph might be disconnected for k = 3, yet, we give upper and lower bounds on the flip distance from one order-k triangulation to another in certain settings. Later, we show that there exists a length-decreasing sequence of plane spanning trees of S that converges to the minimum spanning tree of S with respect to an arbitrary convex distance function. Each pair of consecutive trees in the sequence is contained in a constrained convex shape Delaunay graph. In addition, we give a linear upper bound and specific bounds when the convex shape is a square. With focus still on convex distance functions, we study Hamiltonicity in k-order convex shape Delaunay graphs. Depending on the convex shape, we provide several upper bounds for the minimum k for which the k-order convex shape Delaunay graph is always Hamiltonian. In addition, we provide lower bounds when the convex shape is in a set of certain regular polygons. Finally, we revisit an affine invariant triangulation, which is a special type of convex shape Delaunay triangulation. We show that many properties of the standard Delaunay triangulations carry over to these triangulations. Also, motivated by this affine invariant triangulation, we study different triangulation methods for producing other affine invariant geometric objects.Esta tesis estudia diferentes generalizaciones de la triangulación de Delaunay, tanto desde un punto de vista combinatorio como algorítmico. La triangulación de Delaunay de un conjunto de puntos S, denotada DT(S), tiene como conjunto de vértices a S. Una arista uv está en DT(S) si satisface la propiedad del círculo vacío: existe un círculo con u y v en su frontera que no contiene ningún punto de S en su interior. Debido a distintos criterios de optimización, se han propuesto varias generalizaciones de la DT (S). Hoy en día, se conocen bastantes propiedades de la DT(S), sin embargo, poco se sabe sobre sus generalizaciones. La pregunta principal que exploramos es: ¿Hasta qué punto las propiedades de la DT(S) se pueden extender para generalizaciones de gráficas de Delaunay? Primero, exploramos la conectividad de la gráfica de flips de las triangulaciones de Delaunay de orden alto de un conjunto de puntos S en el plano. La gráfica de flips de triangulaciones de orden k = 3 podría ser disconexa, sin embargo, nosotros damos una cota superior e inferior para la distancia en flips de una triangulación de orden k a alguna otra cuando S cumple con ciertas características. Luego, probamos que existe una secuencia de árboles generadores sin cruces tal que la suma total de la longitud de las aristas con respecto a una distancia convexa arbitraria es decreciente y converge al árbol generador mínimo con respecto a la distancia correspondiente. Cada par de árboles consecutivos en la secuencia se encuentran en una triangulación de Delaunay con restricciones. Adicionalmente, damos una cota superior lineal para la longitud de la secuencia y cotas específicas cuando el conjunto convexo es un cuadrado. Aún concentrados en distancias convexas, estudiamos hamiltonicidad en las gráficas de Delaunay de distancia convexa de k-orden. Dependiendo en la distancia convexa, exhibimos diversas cotas superiores para el mínimo valor de k que satisface que la gráfica de Delaunay de distancia convexa de orden-k es hamiltoniana. También damos cotas inferiores para k cuando el conjunto convexo pertenece a un conjunto de ciertos polígonos regulares. Finalmente, re-visitamos una triangulación afín invariante, la cual es un caso especial de triangulación de Delaunay de distancia convexa. Probamos que muchas propiedades de la triangulación de Delaunay estándar se preservan en estas triangulaciones. Además, motivados por esta triangulación afín invariante, estudiamos diferentes algoritmos que producen otros objetos geométricos afín invariantes