Bichromatic compatible matchings

Aloupis, Greg; Barba, Luis; Langerman, Stefan; Souvaine, Diane L.

doi:10.1016/j.comgeo.2014.08.009

Cited by 14 publications

(3 citation statements)

References 13 publications

Supporting

Mentioning

Contrasting

Order By: Relevance

“…Considering the notion of compatibility, most of the work has been done in the straight-line setting, e.g., in the context of perfect matchings with [5,7] or without [1,2] vertex coloring, or for edge-disjoint compatibility [3,11]. Aichholzer et al [4] showed, in the straight-line setting, that the compatibility graph of plane spanning trees is connected with diameter O(log k), where k denotes the number of convex layers of the point set.…”

Section: Related Workmentioning

confidence: 99%

Plane Spanning Trees in Edge-Colored Simple Drawings of $$K_{n}$$

Aichholzer

Hoffmann

Obenaus

et al. 2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

For a simple drawing D of the complete graph Kn, two (plane) subdrawings are compatible if their union is plane. Let TD be the set of all plane spanning trees on D and F(TD) be the compatibility graph that has a vertex for each element in TD and two vertices are adjacent if and only if the corresponding trees are compatible. We show, on the one hand, that F(TD) is connected if D is a cylindrical, monotone, or strongly c-monotone drawing. On the other hand, we show that the subgraph of F(TD) induced by stars, double stars, and twin stars is also connected. In all cases the diameter of the corresponding compatibility graph is at most linear in n.

show abstract

Section: Related Workmentioning

confidence: 99%

Plane Spanning Trees in Edge-Colored Simple Drawings of $$K_{n}$$

Aichholzer

Hoffmann

Obenaus

et al. 2020

Lecture Notes in Computer Science

View full text Add to dashboard Cite

show abstract

“…[3] proved that such a sequence of at most O(log n) steps always exists. Questions of whether any matching of a given point set can be transformed into any other and how many steps it takes (that is, the connectivity of and the distance in the so-called reconfiguration graph of matchings, as well as its other properties) have been investigated also for matchings on bicolored point sets and for edge-disjoint compatible matchings, see for example [5,8,19]. Our results.…”

Section: Introductionmentioning

confidence: 97%

On Compatible Matchings

Aichholzer,

Arroyo,

Masárová

et al. 2021

Preprint

View full text Add to dashboard Cite

A matching is compatible to two or more labeled point sets of size n with labels {1, . . . , n} if its straight-line drawing on each of these point sets is crossing-free. We study the maximum number of edges in a matching compatible to two or more labeled point sets in general position in the plane. We show that for any two labeled convex sets of n points there exists a compatible matching with √ 2n edges. More generally, for any labeled point sets we construct compatible matchings of size Ω(n 1/ ). As a corresponding upper bound, we use probabilistic arguments to show that for any given sets of n points there exists a labeling of each set such that the largest compatible matching has O(n 2/( +1) ) edges. Finally, we show that Θ(log n) copies of any set of n points are necessary and sufficient for the existence of a labeling such that any compatible matching consists only of a single edge.

show abstract

“…Flips in triangulations are an instance of a more general question: Given a class of spanning graphs G of a point set S, which local operations can transform one element of G into another? Several operators have been introduced for different classes of spanning graphs that are not triangulations, such as perfect matchings, plane Hamiltonian cycles and plane spanning trees [6,9,36]. One class of graphs that has received much attention in these type of operators, is the set of plane spanning trees of a point set.…”

Section: Convex Shape Delaunay Graphs and Treesmentioning

confidence: 99%

Generalized Delaunay triangulations : graph-theoretic properties and algorithms

Cano Vila

View full text Add to dashboard Cite

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.

show abstract

Bichromatic compatible matchings

Cited by 14 publications

References 13 publications

Plane Spanning Trees in Edge-Colored Simple Drawings of $$K_{n}$$

Plane Spanning Trees in Edge-Colored Simple Drawings of $$K_{n}$$

On Compatible Matchings

Generalized Delaunay triangulations : graph-theoretic properties and algorithms

Contact Info

Product

Resources

About