arXiv subjects

Carlos Seara

Publications and source records attributed to Carlos Seara.

At least 19 recordsLinked to original sources

arXiv2026

A Geometric View of Combinatorial Fiedler Theory

Recently, Andrade and Dahl introduced combinatorial Fiedler theory by studying a parameter $b(G)$ defined as the $\ell_1$-analog of the Rayleigh quotient minimization characterization of the algebraic connectivity of a graph $G=(V,E)$. In this work, we study the corresponding maximization problem, which plays the role of the $\ell_1$-analog of the largest Laplacian eigenvalue. We show that the new parameter $B(G)$ associated with this maximization problem admits a simple exact description: it is the average of the two largest vertex degrees of $G$. A unified combinatorial treatment of the minimization and maximization problems is presented first. Later, both optimization problems are reinterpreted in a geometrical setting. The feasible set is identified with a $(n-2)$-dimensional cuboctahedron shell where $n=|V|$. Additional structure is presented for this polyhedron, including the fact that maximizing solutions arise at its vertices and minimizing solutions arise at the centers of its facets. Finally, we analyze the number of optimal vectors for $b(G)$ and $B(G)$ for several graph families. Although the value of $B(G)$ is determined by the two largest degrees, we prove that counting the vectors that attain this value is actually $\#\mathrm{P}$-complete.

cs.CG↗

arXiv2026

Maximum-Weight Two Boxes Symmetric Difference Problem

Let $P$ be a set of $n$ points in the plane, where each element of $P$ is assigned a weight $\omega(p)$, positive or negative. In this paper, we present an algorithm that runs in $O(n^4\log n)$ time and $O(n)$ space to find two possibly overlapping axis-aligned rectangles $A$ and $B$ so as to maximize the total weight of the points contained in the symmetric difference of $A$ and $B$. The same optimization framework can easily be adapted to solve related problems such as to maximize the total weight in the symmetric difference of $k \geq 3$ boxes and/or in the union of $k \geq 2$ boxes.

cs.CG↗

arXiv2023

The Voronoi Diagram of Rotating Rays with applications to Floodlight Illumination

We study the Voronoi Diagram of Rotating Rays, a Voronoi structure where the input sites are rays and the distance function between a point and a site/ray, is the counterclockwise angular distance. This novel Voronoi diagram is motivated by illumination or coverage problems, where a domain must be covered by floodlights/wedges of uniform angle, and the goal is to find the minimum angle necessary to cover the domain. We study the diagram in the plane, and we present structural properties, combinatorial complexity bounds, and a construction algorithm. If the rays are induced by a convex polygon, we show how to construct the Voronoi diagram within this polygon in linear time. Using this information, we can find in optimal linear time the Brocard angle, the minimum angle required to illuminate a convex polygon with floodlights of uniform angle.

cs.CG↗

arXiv2023

Center of maximum-sum matchings of bichromatic points

Let $R$ and $B$ be two disjoint point sets in the plane with $|R|=|B|=n$. Let $\mathcal{M}=\{(r_i,b_i),i=1,2,\ldots,n\}$ be a perfect matching that matches points of $R$ with points of $B$ and maximizes $\sum_{i=1}^n\|r_i-b_i\|$, the total Euclidean distance of the matched pairs. In this paper, we prove that there exists a point $o$ of the plane (the center of $\mathcal{M}$) such that $\|r_i-o\|+\|b_i-o\|\le \sqrt{2}~\|r_i-b_i\|$ for all $i\in\{1,2,\ldots,n\}$.

math.CO↗

arXiv2022

Rectilinear Convex Hull of Points in 3D

Let $P$ be a set of $n$ points in $\mathbb{R}^3$ in general position, and let $RCH(P)$ be the rectilinear convex hull of $P$. In this paper we obtain an optimal $O(n\log n)$-time and $O(n)$-space algorithm to compute $RCH(P)$. We also obtain an efficient $O(n\log^2 n)$-time and $O(n\log n)$-space algorithm to compute and maintain the set of vertices of the rectilinear convex hull of $P$ as we rotate $\mathbb R^3$ around the $z$-axis. Finally we study some properties of the rectilinear convex hulls of point sets in $\mathbb{R}^3$.

cs.CG↗

arXiv2022

Separating bichromatic point sets in the plane by restricted orientation convex hulls

We explore the separability of point sets in the plane by a restricted-orientation convex hull, which is an orientation-dependent, possibly disconnected, and non-convex enclosing shape that generalizes the convex hull. Let $R$ and $B$ be two disjoint sets of red and blue points in the plane, and $\mathcal{O}$ be a set of $k \geq 2$ lines passing through the origin. We study the problem of computing the set of orientations of the lines of $\mathcal{O}$ for which the $\mathcal{O}$-convex hull of $R$ contains no points of $B$. For $k=2$ orthogonal lines we have the rectilinear convex hull. In optimal $O(n \log n)$ time and $O(n)$ space, $n = \vert R \vert + \vert B \vert$, we compute the set of rotation angles such that, after simultaneously rotating the lines of $\mathcal{O}$ around the origin in the same direction, the rectilinear convex hull of $R$ contains no points of $B$. We generalize this result to the case where $\mathcal{O}$ is formed by $k \geq 2$ lines with arbitrary orientations. In the counter-clockwise circular order of the lines of $\mathcal{O}$, let $α_i$ be the angle required to clockwise rotate the $i$th line so it coincides with its successor. We solve the problem in this case in $O(1/Θ\cdot N \log N)$ time and $O(1/Θ\cdot N)$ space, where $Θ= \min \{ α_1,\ldots,α_k \}$ and $N=\max\{k,\vert R \vert + \vert B \vert \}$. We finally consider the case in which $\mathcal{O}$ is formed by $k=2$ lines, one of the lines is fixed, and the second line rotates by an angle that goes from $0$ to $π$. We show that this last case can also be solved in optimal $O(n\log n)$ time and $O(n)$ space, where $n = \vert R \vert + \vert B \vert$.

cs.CG↗

arXiv2020

Shortest Watchman Tours in Simple Polygons under Rotated Monotone Visibility

We present an $O(nrG)$ time algorithm for computing and maintaining a minimum length shortest watchman tour that sees a simple polygon under monotone visibility in direction $θ$, while $θ$ varies in $[0,180^{\circ})$, obtaining the directions for the tour to be the shortest one over all tours, where $n$ is the number of vertices, $r$ is the number of reflex vertices, and $G\leq r$ is the maximum number of gates of the polygon used at any time in the algorithm.

cs.CG↗

arXiv2019

On Maximum-Sum Matchings of Points

Huemer et al. (Discrete Mathematics, 2019) proved that for any two point sets $R$ and $B$ with $|R|=|B|$, the perfect matching that matches points of $R$ with points of $B$, and maximizes the total \emph{squared} Euclidean distance of the matched pairs, verifies that all the disks induced by the matching have a common point. Each pair of matched points $p\in R$ and $q\in B$ induces the disk of smallest diameter that covers $p$ and $q$. Following this research line, in this paper we consider the perfect matching that maximizes the total Euclidean distance. First, we prove that this new matching for $R$ and $B$ does not always ensure the common intersection property of the disks. Second, we extend the study of this new matching for sets of $2n$ uncolored points in the plane, where a matching is just a partition of the points into $n$ pairs. As the main result, we prove that in this case all disks of the matching do have a common point. This implies a big improvement on a conjecture of Andy Fingerhut in 1995, about a maximum matching of $2n$ points in the plane.

cs.CG↗

arXiv2019

Maximum rectilinear convex subsets

Let $P$ be a set of $n$ points in the plane. We consider a variation of the classical Erd\H{o}s-Szekeres problem, presenting efficient algorithms with $O(n^3)$ running time and $O(n^2)$ space complexity that compute: (1) A subset $S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the maximum number of points from $P$, (2) a subset $S$ of $P$ such that the boundary of the rectilinear convex hull of $S$ has the maximum number of points from $P$ and its interior contains no element of $P$, (3) a subset $S$ of $P$ such that the rectilinear convex hull of $S$ has maximum area and its interior contains no element of $P$, and (4) when each point of $P$ is assigned a weight, positive or negative, a subset $S$ of $P$ that maximizes the total weight of the points in the rectilinear convex hull of $S$. We also revisit the problems of computing a maximum-area orthoconvex polygon and computing a maximum-area staircase polygon, amidst a point set in a rectangular domain. We obtain new and simpler algorithms to solve both problems with the same complexity as in the state of the art.

cs.CG↗

arXiv2019

Trees whose even-degree vertices induce a path are antimagic

An antimagic labeling a connected graph $G$ is a bijection from the set of edges $E(G)$ to $\{1,2,\dots,|E(G)|\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $v$ is the sum of the labels assigned to edges incident to $v$. A graph is called antimagic if it has an antimagic labeling. In 1990, Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic; however, the conjecture remains open, even for trees. In this note we prove that trees whose vertices of even degree induce a path are antimagic, extending a result given by Liang, Wong, and Zhu [Discrete Math. 331 (2014) 9--14].

math.CO↗

arXiv2019

Caterpillars are Antimagic

An antimagic labeling of a graph $G$ is an injection from $E(G)$ to $\{1,2,\dots,|E(G)|\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $u$ is the sum of the labels assigned to edges incident to $u$. A graph is called antimagic when it has an antimagic labeling. Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic and the conjecture remains open even for trees. Here we prove that caterpillars are antimagic by means of an $O(n \log n)$ algorithm.

math.CO↗

arXiv2019

Matching points with disks with a common intersection

We consider matchings with diametral disks between two sets of points R and B. More precisely, for each pair of matched points p in R and q in B, we consider the disk through p and q with the smallest diameter. We prove that for any R and B such that |R|=|B|, there exists a perfect matching such that the diametral disks of the matched point pairs have a common intersection. In fact, our result is stronger, and shows that a maximum weight perfect matching has this property.

cs.CG↗

arXiv2019

$K_{1,3}$-covering red and blue points in the plane

We say that a finite set of red and blue points in the plane in general position can be $K_{1,3}$-covered if the set can be partitioned into subsets of size $4$, with $3$ points of one color and $1$ point of the other color, in such a way that, if at each subset the fourth point is connected by straight-line segments to the same-colored points, then the resulting set of all segments has no crossings. We consider the following problem: Given a set $R$ of $r$ red points and a set $B$ of $b$ blue points in the plane in general position, how many points of $R\cup B$ can be $K_{1,3}$-covered? and we prove the following results: (1) If $r=3g+h$ and $b=3h+g$, for some non-negative integers $g$ and $h$, then there are point sets $R\cup B$, like $\{1,3\}$-equitable sets (i.e., $r=3b$ or $b=3r$) and linearly separable sets, that can be $K_{1,3}$-covered. (2) If $r=3g+h$, $b=3h+g$ and the points in $R\cup B$ are in convex position, then at least $r+b-4$ points can be $K_{1,3}$-covered, and this bound is tight. (3) There are arbitrarily large point sets $R\cup B$ in general position, with $r=b+1$, such that at most $r+b-5$ points can be $K_{1,3}$-covered. (4) If $b\le r\le 3b$, then at least $\frac{8}{9}(r+b-8)$ points of $R\cup B$ can be $K_{1,3}$-covered. For $r>3b$, there are too many red points and at least $r-3b$ of them will remain uncovered in any $K_{1,3}$-covering. Furthermore, in all the cases we provide efficient algorithms to compute the corresponding coverings.

math.CO↗

arXiv2019

Antimagic Labelings of Caterpillars

A $k$-antimagic labeling of a graph $G$ is an injection from $E(G)$ to $\{1,2,\dots,|E(G)|+k\}$ such that all vertex sums are pairwise distinct, where the vertex sum at vertex $u$ is the sum of the labels assigned to edges incident to $u$. We call a graph $k$-antimagic when it has a $k$-antimagic labeling, and antimagic when it is 0-antimagic. Hartsfield and Ringel conjectured that every simple connected graph other than $K_2$ is antimagic, but the conjecture is still open even for trees. Here we study $k$-antimagic labelings of caterpillars, which are defined as trees the removal of whose leaves produces a path, called its spine. As a general result, we use constructive techniques to prove that any caterpillar of order $n$ is $(\lfloor (n-1)/2 \rfloor - 2)$-antimagic. Furthermore, if $C$ is a caterpillar with a spine of order $s$, we prove that when $C$ has at least $\lfloor (3s+1)/2 \rfloor$ leaves or $\lfloor (s-1)/2 \rfloor$ consecutive vertices of degree at most 2 at one end of a longest path, then $C$ is antimagic. As a consequence of a result by Wong and Zhu, we also prove that if $p$ is a prime number, any caterpillar with a spine of order $p$, $p-1$ or $p-2$ is $1$-antimagic.

math.CO↗

arXiv2018

Capturing points with a rotating polygon (and a 3D extension)

We study the problem of rotating a simple polygon to contain the maximum number of elements from a given point set in the plane. We consider variations of this problem where the rotation center is a given point or lies on a line segment, a line, or a polygonal chain. We also solve an extension to 3D where we rotate a polyhedron around a given point to contain the maximum number of elements from a set of points in the space.

cs.CG↗

arXiv2018

Optimizing generalized kernels of polygons

Let $\mathcal{O}$ be a set of $k$ orientations in the plane, and let $P$ be a simple polygon in the plane. Given two points $p,q$ inside $P$, we say that $p$ $\mathcal{O}$-\emph{sees} $q$ if there is an $\mathcal{O}$-\emph{staircase} contained in $P$ that connects $p$ and~$q$. The \emph{$\mathcal{O}$-Kernel} of the polygon $P$, denoted by $\mathcal{O}$-$\rm kernel(P)$, is the subset of points of $P$ which $\mathcal{O}$-see all the other points in $P$. This work initiates the study of the computation and maintenance of $\mathcal{O}$-$\rm kernel(P)$ as we rotate the set $\mathcal{O}$ by an angle $\theta$, denoted by $\mathcal{O}$-$\rm kernel_{\theta}(P)$. In particular, we consider the case when the set $\mathcal{O}$ is formed by either one or two orthogonal orientations, $\mathcal{O}=\{0^\circ\}$ or $\mathcal{O}=\{0^\circ,90^\circ\}$. For these cases and $P$ being a simple polygon, we design efficient algorithms for computing the $\mathcal{O}$-$\rm kernel_{\theta}(P)$ while $\theta$ varies in $[-\frac{\pi}{2},\frac{\pi}{2})$, obtaining: (i)~the intervals of angle~$\theta$ where $\mathcal{O}$-$\rm kernel_{\theta}(P)$ is not empty, (ii)~a value of angle~$\theta$ where $\mathcal{O}$-$\rm kernel_{\theta}(P)$ optimizes area or perimeter. Further, we show how the algorithms can be improved when $P$ is a simple orthogonal polygon. In addition, our results are extended to the case of a set $\mathcal{O}=\{\alpha_1,\dots,\alpha_k\}$.

cs.CG↗

arXiv2017

Rectilinear Convex Hull with minimum area

Let $P$ be a planar set of $n$ points in general position. We consider the problem of computing an orientation of the plane for which the Rectilinear Convex Hull of $P$ has minimum area. Bae et al. (Computational Geometry: Theory and Applications, Vol. 42, 2009) solved the problem in quadratic time and linear space. We describe an algorithm that reduces this time complexity to $Θ(n \log n)$.

cs.CG↗

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↗