351 research outputs found
Compatible 4-Holes in Point Sets
Counting interior-disjoint empty convex polygons in a point set is a typical
Erd\H{o}s-Szekeres-type problem. We study this problem for 4-gons. Let be a
set of points in the plane and in general position. A subset of ,
with four points, is called a -hole in if is in convex position and
its convex hull does not contain any point of in its interior. Two 4-holes
in are compatible if their interiors are disjoint. We show that
contains at least pairwise compatible 4-holes.
This improves the lower bound of which is implied by a
result of Sakai and Urrutia (2007).Comment: 17 page
Higher-Order Triangular-Distance Delaunay Graphs: Graph-Theoretical Properties
We consider an extension of the triangular-distance Delaunay graphs
(TD-Delaunay) on a set of points in the plane. In TD-Delaunay, the convex
distance is defined by a fixed-oriented equilateral triangle ,
and there is an edge between two points in if and only if there is an empty
homothet of having the two points on its boundary. We consider
higher-order triangular-distance Delaunay graphs, namely -TD, which contains
an edge between two points if the interior of the homothet of
having the two points on its boundary contains at most points of . We
consider the connectivity, Hamiltonicity and perfect-matching admissibility of
-TD. Finally we consider the problem of blocking the edges of -TD.Comment: 20 page
Packing Plane Perfect Matchings into a Point Set
Given a set of points in the plane, where is even, we consider
the following question: How many plane perfect matchings can be packed into
? We prove that at least plane perfect matchings
can be packed into any point set . For some special configurations of point
sets, we give the exact answer. We also consider some extensions of this
problem
- β¦