10-Gabriel graphs are Hamiltonian

Given a set $S$ of points in the plane, the $k$-Gabriel graph of $S$ is the geometric graph with vertex set $S$, where $p_i,p_j\in S$ are connected by an edge if and only if the closed disk having segment $\bar{p_ip_j}$ as diameter contains at most $k$ points of $S \setminus \{p_i,p_j\}$. We consider the following question: What is the minimum value of $k$ such that the $k$-Gabriel graph of every point set $S$ contains a Hamiltonian cycle? For this value, we give an upper bound of 10 and a lower bound of 2. The best previously known values were 15 and 1, respectively

Flip Graphs of Degree-Bounded (Pseudo-)Triangulations

We study flip graphs of triangulations whose maximum vertex degree is bounded by a constant $k$. In particular, we consider triangulations of sets of $n$ points in convex position in the plane and prove that their flip graph is connected if and only if $k > 6$; the diameter of the flip graph is $O(n^2)$. We also show that, for general point sets, flip graphs of pointed pseudo-triangulations can be disconnected for $k \leq 9$, and flip graphs of triangulations can be disconnected for any $k$. Additionally, we consider a relaxed version of the original problem. We allow the violation of the degree bound $k$ by a small constant. Any two triangulations with maximum degree at most $k$ of a convex point set are connected in the flip graph by a path of length $O(n \log n)$, where every intermediate triangulation has maximum degree at most $k+4$.Comment: 13 pages, 12 figures, acknowledgments update

Higher-Order Triangular-Distance Delaunay Graphs: Graph-Theoretical Properties

We consider an extension of the triangular-distance Delaunay graphs (TD-Delaunay) on a set $P$ of points in the plane. In TD-Delaunay, the convex distance is defined by a fixed-oriented equilateral triangle $\triangledown$, and there is an edge between two points in $P$ if and only if there is an empty homothet of $\triangledown$ having the two points on its boundary. We consider higher-order triangular-distance Delaunay graphs, namely $k$-TD, which contains an edge between two points if the interior of the homothet of $\triangledown$ having the two points on its boundary contains at most $k$ points of $P$. We consider the connectivity, Hamiltonicity and perfect-matching admissibility of $k$-TD. Finally we consider the problem of blocking the edges of $k$-TD.Comment: 20 page

On structural and graph theoretic properties of higher order delaunay graphs

Given a set P of n points in the plane, the order-k Delaunay graph is a graph with vertex set P and an edge exists between two points p, q ∈ P when there is a circle through p and q with at most k other points of P in its interior. We provide upper and lower bounds on the number of edges in an order-k Delaunay graph. We study the combinatorial structure of the set of triangulations that can be constructed with edges of this graph. Furthermore, we show that the order-k Delaunay graph is connected under the flip operation when k ≤ 1 but not necessarily connected for other values of k. If P is in convex position then the order-k Delaunay graph is connected for all k < 0. We show that the order-k Gabriel graph, a subgraph of the order-k Delaunay graph, is Hamiltonian for k ≥ 15. Finally, the order-k Delaunay graph can be used to efficiently solve a coloring problem with applications to frequency assignments in cellular networks

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