Geometria discreta i computacional. Problemes

2006/200

Geometria discreta i computacional. Problemes

2009/201

Witness gabriel graphs

We consider a generalization of the Gabriel graph, the witness Gabriel graph. Given a set of vertices P and a set of witness points W in the plane, there is an edge ab between two points of P in the witness Gabriel graph $GG^-$(P,W) if and only if the closed disk with diameter ab does not contain any witness point (besides possibly a and/or b). We study several properties of the witness Gabriel graph, both as a proximity graph and as a new tool in graph drawing.Postprint (published version

Stabbing simplices of point sets with k-flats

Let S be a set of n points inRdin general position.A set H of k-flats is called an mk-stabber of S if the relative interior of anym-simplex with vertices in S is intersected by at least one element of H. In thispaper we give lower and upper bounds on the size of mínimum mk-stabbers of point sets in Rd. We study mainly mk-stabbers in the plane and in R3Peer ReviewedPostprint (published version

Witness bar visibility graphs

Bar visibility graphs were introduced in the seventies as a model for some VLSI layout problems. They have been also studied since then by the graph drawing community, and recently several generalizations and restricted versions have been proposed. We introduce a generalization, witness-bar visibility graphs, and we prove that this class encom- passes all the bar-visibility variations considered so far. In addition, we show that many classes of graphs are contained in this family, including in particular all planar graphs, interval graphs, circular arc graphs and permutation graphsPeer ReviewedPostprint (published version

Connectivity-preserving transformations of binary images

A binary image \emph{I} is $B_a$, $W_b$-connected, where \emph{a,b} ∊ {4,8}, if its foreground is \emph{a}-connected and its background is \emph{b}-connected. We consider a local modification of a $B_a$, $W_b$-connected image \emph{I} in which a black pixel can be interchanged with an adjacent white pixel provided that this preserves the connectivity of both the foreground and the background of \emph{I}. We have shown that for any (\emph{a,b}) ∊ {(4,8),(8,4),(8,8)}, any two $B_a$, $W_b$-connected images \emph{I} and \emph{J} each with n black pixels differ by a sequence of $\theta(n^2)$ interchanges. We have also shown that any two $B_4$, $W_4$-connected images \emph{I} and \emph{J} each with n black pixels differ by a sequence of O($n^4$) interchanges.Postprint (published version

Compatible matchings in geometric graphs

Two non-crossing geometric graphs on the same set of points are compatible if their union is also non-crossing. In this paper, we prove that every graph G that has an outerplanar embedding admits a non-crossing perfect matching compatible with G. Moreover, for non-crossing geometric trees and simple polygons, we study bounds on the minimum number of edges that a compatible non-crossing perfect matching must share with the tree or the polygon. We also give bounds on the maximal size of a compatible matching (not necessarily perfect) that is disjoint from the tree or the polygon.Postprint (published version

Graphs of non-crossing perfect matchings

Let Pn be a set of n = 2m points that are the vertices of a convex polygon, and let Mm be the graph having as vertices all the perfect matchings in the point set Pn whose edges are straight line segments and do not cross, and edges joining two perfect matchings M1 and M2 if M2 = M1 ¡ (a; b) ¡ (c; d) + (a; d) + (b; c) for some points a; b; c; d of Pn. We prove the following results about Mm: its diameter is m ¡ 1; it is bipartite for every m; the connectivity is equal to m ¡ 1; it has no Hamilton path for m odd, m > 3; and finally it has a Hamilton cycle for every m even, m>=4

On some partitioning problems for two-colored point sets

Let S be a two-colored set of n points in general position in the plane. We show that S admits at least 2 n 17 pairwise disjoint monochromatic triangles with vertices in S and empty of points of S. We further show that S can be partitioned into 3 n 11 subsets with pairwise disjoint convex hull such that within each subset all but at most one point have the same color. A lower bound on the number of subsets needed in any such partition is also given.Postprint (published version

On polygons enclosing point sets II

Let R and B be disjoint point sets such that $R\cup B$ is in general position. We say that B encloses by R if there is a simple polygon P with vertex set B such that all the elements in R belong to the interior of P. In this paper we prove that if the vertices of the convex hull of $R\cup B$ belong to B, and |R| ≤ |Conv(B)| − 1 then B encloses R. The bound is tight. This improves on results of a previous paper in which it was proved that if |R| ≤ 56|Conv (B)| then B encloses R. To obtain our result we prove the next result which is interesting on its own right: Let P be a convex polygon with n vertices \emph{p_1},...,\emph{p_n} and S a set of m points contained in the interior of P, m ≤ n−1. Then there is a convex decomposition {$P_1$,...,$P_n$} of P such that all points from S lie on the boundaries of $P_1$,...,$P_n$, and each $P_i$ contains a whole edge of P on its boundary.Postprint (published version