arXiv subjects

Carlos Seara

Publications and source records attributed to Carlos Seara.

25 records · Page 2Linked to original sources

arXiv2017

On the $O_β$-hull of a planar point set

We study the $O_β$-hull of a planar point set, a generalization of the Orthogonal Convex Hull where the coordinate axes form an angle $β$. Given a set $P$ of $n$ points in the plane, we show how to maintain the $O_β$-hull of $P$ while $β$ runs from $0$ to $π$ in $O(n \log n)$ time and $O(n)$ space. With the same complexity, we also find the values of $β$ that maximize the area and the perimeter of the $O_β$-hull and, furthermore, we find the value of $β$ achieving the best fitting of the point set $P$ with a two-joint chain of alternate interior angle $β$.

cs.CG↗

arXiv2017

Balanced partitions of 3-colored geometric sets in the plane

Let $S$ be a finite set of geometric objects partitioned into classes or \emph{colors}. A subset $S'\subseteq S$ is said to be \emph{balanced} if $S'$ contains the same amount of elements of $S$ from each of the colors. We study several problems on partitioning $3$-colored sets of points and lines in the plane into two balanced subsets: (a) We prove that for every 3-colored arrangement of lines there exists a segment that intersects exactly one line of each color, and that when there are $2m$ lines of each color, there is a segment intercepting $m$ lines of each color. (b) Given $n$ red points, $n$ blue points and $n$ green points on any closed Jordan curve $γ$, we show that for every integer $k$ with $0 \leq k \leq n$ there is a pair of disjoint intervals on $γ$ whose union contains exactly $k$ points of each color. (c) Given a set $S$ of $n$ red points, $n$ blue points and $n$ green points in the integer lattice satisfying certain constraints, there exist two rays with common apex, one vertical and one horizontal, whose union splits the plane into two regions, each one containing a balanced subset of $S$.

cs.CG↗

arXiv2017

On Hamiltonian alternating cycles and paths

We undertake a study on computing Hamiltonian alternating cycles and paths on bicolored point sets. This has been an intensively studied problem, not always with a solution, when the paths and cycles are also required to be plane. In this paper, we relax the constraint on the cycles and paths from being plane to being 1-plane, and deal with the same type of questions as those for the plane case, obtaining a remarkable variety of results. Among them, we prove that a 1-plane Hamiltonian alternating cycle on a bicolored point set in general position can always be obtained, and that when the point set is in convex position, every Hamiltonian alternating cycle with minimum number of crossings is 1-plane. Further, for point sets in convex position, we provide $O(n)$ and $O(n^2)$ time algorithms for computing, respectively, Hamiltonian alternating cycles and paths with minimum number of crossings.

cs.CG↗

arXiv2017

Stabbing segments with rectilinear objects

Given a set $S$ of $n$ line segments in the plane, we say that a region $\mathcal{R}\subseteq \mathbb{R}^2$ is a {\em stabber} for $S$ if $\mathcal{R}$ contains exactly one endpoint of each segment of $S$. In this paper we provide optimal or near-optimal algorithms for reporting all combinatorially different stabbers for several shapes of stabbers. Specifically, we consider the case in which the stabber can be described as the intersection of axis-parallel halfplanes (thus the stabbers are halfplanes, strips, quadrants, $3$-sided rectangles, or rectangles). The running times are $O(n)$ (for the halfplane case), $O(n\log n)$ (for strips, quadrants, and 3-sided rectangles), and $O(n^2 \log n)$ (for rectangles).

cs.CG↗

arXiv2014

New results on stabbing segments with a polygon

We consider a natural variation of the concept of stabbing a segment by a simple polygon: a segment is stabbed by a simple polygon $\mathcal{P}$ if at least one of its two endpoints is contained in $\mathcal{P}$. A segment set $S$ is stabbed by $\mathcal{P}$ if every segment of $S$ is stabbed by $\mathcal{P}$. We show that if $S$ is a set of pairwise disjoint segments, the problem of computing the minimum perimeter polygon stabbing $S$ can be solved in polynomial time. We also prove that for general segments the problem is NP-hard. Further, an adaptation of our polynomial-time algorithm solves an open problem posed by Löffler and van Kreveld [Algorithmica 56(2), 236--269 (2010)] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments.

cs.CC↗

arXiv2007

Extremal Graph Theory for Metric Dimension and Diameter

A set of vertices $S$ \emph{resolves} a connected graph $G$ if every vertex is uniquely determined by its vector of distances to the vertices in $S$. The \emph{metric dimension} of $G$ is the minimum cardinality of a resolving set of $G$. Let $\mathcal{G}_{β,D}$ be the set of graphs with metric dimension $β$ and diameter $D$. It is well-known that the minimum order of a graph in $\mathcal{G}_{β,D}$ is exactly $β+D$. The first contribution of this paper is to characterise the graphs in $\mathcal{G}_{β,D}$ with order $β+D$ for all values of $β$ and $D$. Such a characterisation was previously only known for $D\leq2$ or $β\leq1$. The second contribution is to determine the maximum order of a graph in $\mathcal{G}_{β,D}$ for all values of $D$ and $β$. Only a weak upper bound was previously known.

math.CO↗

arXiv2006

On the Metric Dimension of Cartesian Products of Graphs

A set S of vertices in a graph G resolves G if every vertex is uniquely determined by its vector of distances to the vertices in S. The metric dimension of G is the minimum cardinality of a resolving set of G. This paper studies the metric dimension of cartesian products G*H. We prove that the metric dimension of G*G is tied in a strong sense to the minimum order of a so-called doubly resolving set in G. Using bounds on the order of doubly resolving sets, we establish bounds on G*H for many examples of G and H. One of our main results is a family of graphs G with bounded metric dimension for which the metric dimension of G*G is unbounded.

math.CO↗