46 research outputs found
On the Connectedness and Diameter of a Geometric Johnson Graph
Let be a set of points in general position in the plane. A subset
of is called an \emph{island} if there exists a convex set such that . In this paper we define the \emph{generalized island Johnson
graph} of as the graph whose vertex consists of all islands of of
cardinality , two of which are adjacent if their intersection consists of
exactly elements. We show that for large enough values of , this graph
is connected, and give upper and lower bounds on its diameter
Balanced Islands in Two Colored Point Sets in the Plane
Let be a set of points in general position in the plane, of which
are red and of which are blue. In this paper we prove that there exist: for
every , a convex set containing
exactly red points and exactly
blue points of ; a convex set containing exactly red points and exactly blue points of . Furthermore, we present
polynomial time algorithms to find these convex sets. In the first case we
provide an time algorithm and an time algorithm in the
second case. Finally, if is
small, that is, not much larger than , we improve the running
time to
Matching random colored points with rectangles
Let S[0,1]2 be a set of n points, randomly and uniformly selected. Let RB be a random partition, or coloring, of S in which each point of S is included in R uniformly at random with probability 1/2. We study the random variable M(n) equal to the number of points of S that are covered by the rectangles of a maximum strong matching of S with axis-aligned rectangles. The matching consists of closed rectangles that cover exactly two points of S of the same color. A matching is strong if all its rectangles are pairwise disjoint. We prove that almost surely M(n)=0.83n for n large enough. Our approach is based on modeling a deterministic greedy matching algorithm, that runs over the random point set, as a Markov chain.Research supported by projects MTM2015-63791-R MINECO/FEDER and Gen.
Cat. DGR 2017SGR1640Postprint (author's final draft
Rainbow polygons for colored point sets in the plane
Given a colored point set in the plane, a perfect rainbow polygon is a simple
polygon that contains exactly one point of each color, either in its interior
or on its boundary. Let denote the smallest size
of a perfect rainbow polygon for a colored point set , and let
be the maximum of
over all -colored point sets in general position; that is, every -colored
point set has a perfect rainbow polygon with at most
vertices. In this paper, we determine the values
of up to , which is the first case where
, and we prove that for , Furthermore, for a -colored set of points in the plane in general
position, a perfect rainbow polygon with at most vertices can be computed in time.Comment: 23 pages, 11 figures, to appear at Discrete Mathematic
Edge-Removal and Non-Crossing Configurations in Geometric Graphs
A geometric graph is a graph G = (V;E) drawn in the plane, such that V is a point set in general position and E is
a set of straight-line segments whose endpoints belong to V . We study the following extremal problem for geometric
graphs: How many arbitrary edges can be removed from a complete geometric graph with n vertices such that the
remaining graph still contains a certain non-crossing subgraph. The non-crossing subgraphs that we consider are
perfect matchings, subtrees of a given size, and triangulations. In each case, we obtain tight bounds on the maximum
number of removable edges.Postprint (published version