Articles de revista
http://hdl.handle.net/2117/3197
Sun, 17 Dec 2017 17:51:25 GMT2017-12-17T17:51:25ZOn Hamiltonian alternating cycles and paths
http://hdl.handle.net/2117/112128
On Hamiltonian alternating cycles and paths
Claverol Aguas, Mercè; García, Alfredo; Garijo Royo, Delia; Seara Ojea, Carlos; Tejel, Javier
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. For point sets in general position, our main result is that it is always possible to obtain a 1-plane Hamiltonian alternating cycle. When the point set is in convex position, we prove that every Hamiltonian alternating cycle with minimum number of crossings is 1-plane, and provide O(n) and O(n2) time algorithms for computing, respectively, Hamiltonian alternating cycles and paths with minimum number of crossings.
Fri, 15 Dec 2017 10:17:05 GMThttp://hdl.handle.net/2117/1121282017-12-15T10:17:05ZClaverol Aguas, MercèGarcía, AlfredoGarijo Royo, DeliaSeara Ojea, CarlosTejel, JavierWe 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. For point sets in general position, our main result is that it is always possible to obtain a 1-plane Hamiltonian alternating cycle. When the point set is in convex position, we prove that every Hamiltonian alternating cycle with minimum number of crossings is 1-plane, and provide O(n) and O(n2) time algorithms for computing, respectively, Hamiltonian alternating cycles and paths with minimum number of crossings.Carathodory's theorem in depth
http://hdl.handle.net/2117/111689
Carathodory's theorem in depth
Fabila Monroy, Ruy; Huemer, Clemens
Let X be a finite set of points in RdRd . The Tukey depth of a point q with respect to X is the minimum number tX(q)tX(q) of points of X in a halfspace containing q. In this paper we prove a depth version of Carathéodory’s theorem. In particular, we prove that there exist a constant c (that depends only on d and tX(q)tX(q) ) and pairwise disjoint sets X1,…,Xd+1¿XX1,…,Xd+1¿X such that the following holds. Each XiXi has at least c|X| points, and for every choice of points xixi in XiXi , q is a convex combination of x1,…,xd+1x1,…,xd+1 . We also prove depth versions of Helly’s and Kirchberger’s theorems.
Mon, 11 Dec 2017 12:19:57 GMThttp://hdl.handle.net/2117/1116892017-12-11T12:19:57ZFabila Monroy, RuyHuemer, ClemensLet X be a finite set of points in RdRd . The Tukey depth of a point q with respect to X is the minimum number tX(q)tX(q) of points of X in a halfspace containing q. In this paper we prove a depth version of Carathéodory’s theorem. In particular, we prove that there exist a constant c (that depends only on d and tX(q)tX(q) ) and pairwise disjoint sets X1,…,Xd+1¿XX1,…,Xd+1¿X such that the following holds. Each XiXi has at least c|X| points, and for every choice of points xixi in XiXi , q is a convex combination of x1,…,xd+1x1,…,xd+1 . We also prove depth versions of Helly’s and Kirchberger’s theorems.Characteristic polynomials of production matrices for geometric graphs
http://hdl.handle.net/2117/111649
Characteristic polynomials of production matrices for geometric graphs
Huemer, Clemens; Pilz, Alexander; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio
An n×n production matrix for a class of geometric graphs has the property that the numbers of these geometric graphs on up to n vertices can be read off from the powers of the matrix. Recently, we obtained such production matrices for non-crossing geometric graphs on point sets in convex position [Huemer, C., A. Pilz, C. Seara, and R.I. Silveira, Production matrices for geometric graphs, Electronic Notes in Discrete Mathematics 54 (2016) 301–306]. In this note, we determine the characteristic polynomials of these matrices. Then, the Cayley-Hamilton theorem implies relations among the numbers of geometric graphs with different numbers of vertices. Further, relations between characteristic polynomials of production matrices for geometric graphs and Fibonacci numbers are revealed.
Mon, 11 Dec 2017 09:53:39 GMThttp://hdl.handle.net/2117/1116492017-12-11T09:53:39ZHuemer, ClemensPilz, AlexanderSeara Ojea, CarlosSilveira, Rodrigo IgnacioAn n×n production matrix for a class of geometric graphs has the property that the numbers of these geometric graphs on up to n vertices can be read off from the powers of the matrix. Recently, we obtained such production matrices for non-crossing geometric graphs on point sets in convex position [Huemer, C., A. Pilz, C. Seara, and R.I. Silveira, Production matrices for geometric graphs, Electronic Notes in Discrete Mathematics 54 (2016) 301–306]. In this note, we determine the characteristic polynomials of these matrices. Then, the Cayley-Hamilton theorem implies relations among the numbers of geometric graphs with different numbers of vertices. Further, relations between characteristic polynomials of production matrices for geometric graphs and Fibonacci numbers are revealed.Internally perfect matroids
http://hdl.handle.net/2117/108462
Internally perfect matroids
Dall, Aaron Matthew
In 1977 Stanley proved that the $h$-vector of a matroid is an $\mathcal{O}$-sequence and conjectured that it is a pure $\mathcal{O}$-sequence. In the subsequent years the validity of this conjecture has been shown for a variety of classes of matroids, though the general case is still open. In this paper we use Las Vergnas' internal order to introduce a new class of matroids which we call internally perfect. We prove that these matroids satisfy Stanley's Conjecture and compare them to other classes of matroids for which the conjecture is known to hold. We also prove that, up to a certain restriction on deletions, every minor of an internally perfect ordered matroid is internally perfect.
Fri, 06 Oct 2017 12:52:55 GMThttp://hdl.handle.net/2117/1084622017-10-06T12:52:55ZDall, Aaron MatthewIn 1977 Stanley proved that the $h$-vector of a matroid is an $\mathcal{O}$-sequence and conjectured that it is a pure $\mathcal{O}$-sequence. In the subsequent years the validity of this conjecture has been shown for a variety of classes of matroids, though the general case is still open. In this paper we use Las Vergnas' internal order to introduce a new class of matroids which we call internally perfect. We prove that these matroids satisfy Stanley's Conjecture and compare them to other classes of matroids for which the conjecture is known to hold. We also prove that, up to a certain restriction on deletions, every minor of an internally perfect ordered matroid is internally perfect.Stabbing segments with rectilinear objects
http://hdl.handle.net/2117/104941
Stabbing segments with rectilinear objects
Claverol Aguas, Mercè; Garijo Royo, Delia; Korman, Matias; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio
Given a set S of n line segments in the plane, we say that a region R¿R2 is a stabber for S if 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(nlog¿n) (for strips, quadrants, and 3-sided rectangles), and O(n2log¿n) (for rectangles).
Fri, 26 May 2017 16:14:19 GMThttp://hdl.handle.net/2117/1049412017-05-26T16:14:19ZClaverol Aguas, MercèGarijo Royo, DeliaKorman, MatiasSeara Ojea, CarlosSilveira, Rodrigo IgnacioGiven a set S of n line segments in the plane, we say that a region R¿R2 is a stabber for S if 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(nlog¿n) (for strips, quadrants, and 3-sided rectangles), and O(n2log¿n) (for rectangles).On perfect and quasiperfect dominations in graphs
http://hdl.handle.net/2117/104244
On perfect and quasiperfect dominations in graphs
Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Cáceres, José; Puertas, M. Luz
A subset S ¿ V in a graph G = ( V , E ) is a k -quasiperfect dominating set (for k = 1) if every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k -quasiperfect dominating set in G is denoted by ¿ 1 k ( G ). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n = ¿ 11 ( G ) = ¿ 12 ( G ) = ... = ¿ 1 ¿ ( G ) = ¿ ( G ) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ¿ 12 ( G ) = ¿ ( G ). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ¿ ( G ).
Wed, 10 May 2017 05:56:12 GMThttp://hdl.handle.net/2117/1042442017-05-10T05:56:12ZHernando Martín, María del CarmenMora Giné, MercèPelayo Melero, Ignacio ManuelCáceres, JoséPuertas, M. LuzA subset S ¿ V in a graph G = ( V , E ) is a k -quasiperfect dominating set (for k = 1) if every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum k -quasiperfect dominating set in G is denoted by ¿ 1 k ( G ). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept and allow us to construct a decreasing chain of quasiperfect dominating numbers n = ¿ 11 ( G ) = ¿ 12 ( G ) = ... = ¿ 1 ¿ ( G ) = ¿ ( G ) in order to indicate how far is G from being perfectly dominated. In this paper we study properties, existence and realization of graphs for which the chain is short, that is, ¿ 12 ( G ) = ¿ ( G ). Among them, one can find cographs, claw-free graphs and graphs with extremal values of ¿ ( G ).Stabbing circles for sets of segments in the plane
http://hdl.handle.net/2117/104128
Stabbing circles for sets of segments in the plane
Claverol Aguas, Mercè; Khramtcova, Elena; Papadopoulou, Evanthia; Saumell, Maria; Seara Ojea, Carlos
Stabbing a set S of n segments in the plane by a line is a well-known problem. In this paper we consider the variation where the stabbing object is a circle instead of a line. We show that the problem is tightly connected to two cluster Voronoi diagrams, in particular, the Hausdorff and the farthest-color Voronoi diagram. Based on these diagrams, we provide a method to compute a representation of all the combinatorially different stabbing circles for S, and the stabbing circles with maximum and minimum radius. We give conditions under which our method is fast. These conditions are satisfied if the segments in S are parallel, resulting in a O(nlog2n) time and O(n) space algorithm. We also observe that the stabbing circle problem for S can be solved in worst-case optimal O(n2) time and space by reducing the problem to computing the stabbing planes for a set of segments in 3D. Finally we show that the problem of computing the stabbing circle of minimum radius for a set of n parallel segments of equal length has an O(nlogn) lower bound.
Fri, 05 May 2017 13:16:02 GMThttp://hdl.handle.net/2117/1041282017-05-05T13:16:02ZClaverol Aguas, MercèKhramtcova, ElenaPapadopoulou, EvanthiaSaumell, MariaSeara Ojea, CarlosStabbing a set S of n segments in the plane by a line is a well-known problem. In this paper we consider the variation where the stabbing object is a circle instead of a line. We show that the problem is tightly connected to two cluster Voronoi diagrams, in particular, the Hausdorff and the farthest-color Voronoi diagram. Based on these diagrams, we provide a method to compute a representation of all the combinatorially different stabbing circles for S, and the stabbing circles with maximum and minimum radius. We give conditions under which our method is fast. These conditions are satisfied if the segments in S are parallel, resulting in a O(nlog2n) time and O(n) space algorithm. We also observe that the stabbing circle problem for S can be solved in worst-case optimal O(n2) time and space by reducing the problem to computing the stabbing planes for a set of segments in 3D. Finally we show that the problem of computing the stabbing circle of minimum radius for a set of n parallel segments of equal length has an O(nlogn) lower bound.General properties of c-circulant digraphs
http://hdl.handle.net/2117/103764
General properties of c-circulant digraphs
Mora Giné, Mercè; Serra Albó, Oriol; Fiol Mora, Miquel Àngel
A digraph is said to be a c-circulant if its adjacency matrix is c-circulant. This paper deals with general properties of this family of digraphs, as isomorphisms, regularity, strong connectivity, diameter and the relation between c-circulant digraphs and the line digraph technique.
Wed, 26 Apr 2017 17:41:13 GMThttp://hdl.handle.net/2117/1037642017-04-26T17:41:13ZMora Giné, MercèSerra Albó, OriolFiol Mora, Miquel ÀngelA digraph is said to be a c-circulant if its adjacency matrix is c-circulant. This paper deals with general properties of this family of digraphs, as isomorphisms, regularity, strong connectivity, diameter and the relation between c-circulant digraphs and the line digraph technique.Production matrices for geometric graphs
http://hdl.handle.net/2117/103649
Production matrices for geometric graphs
Huemer, Clemens; Pilz, Alexander; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio
We present production matrices for non-crossing geometric graphs on point sets in convex position, which allow us to derive formulas for the numbers of such graphs. Several known identities for Catalan numbers, Ballot numbers, and Fibonacci numbers arise in a natural way, and also new formulas are obtained, such as a formula for the number of non-crossing geometric graphs with root vertex of given degree. The characteristic polynomials of some of these production matrices are also presented. The proofs make use of generating trees and Riordan arrays.
Mon, 24 Apr 2017 08:25:14 GMThttp://hdl.handle.net/2117/1036492017-04-24T08:25:14ZHuemer, ClemensPilz, AlexanderSeara Ojea, CarlosSilveira, Rodrigo IgnacioWe present production matrices for non-crossing geometric graphs on point sets in convex position, which allow us to derive formulas for the numbers of such graphs. Several known identities for Catalan numbers, Ballot numbers, and Fibonacci numbers arise in a natural way, and also new formulas are obtained, such as a formula for the number of non-crossing geometric graphs with root vertex of given degree. The characteristic polynomials of some of these production matrices are also presented. The proofs make use of generating trees and Riordan arrays.New results on stabbing segments with a polygon
http://hdl.handle.net/2117/103547
New results on stabbing segments with a polygon
Díaz Bañez, José Miguel; Korman Cozzetti, Matías; Pérez Lantero, Pablo; Pilz, Alexander; Seara Ojea, Carlos; Silveira, Rodrigo Ignacio
We consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that some measure of P (such as area or perimeter) is optimized. We show that if the elements of S are pairwise disjoint, the problem can be solved in polynomial time. In particular, this solves an open problem posed by Loftier and van Kreveld [Algorithmica 56(2), 236-269 (2010)] [16] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments. Our algorithm can also be extended to work for a more general problem, in which instead of segments, the set S consists of a collection of point sets with pairwise disjoint convex hulls. We also prove that for general segments our stabbing problem is NP-hard. (C) 2014 Elsevier B.V. All rights reserved.
Wed, 19 Apr 2017 12:27:10 GMThttp://hdl.handle.net/2117/1035472017-04-19T12:27:10ZDíaz Bañez, José MiguelKorman Cozzetti, MatíasPérez Lantero, PabloPilz, AlexanderSeara Ojea, CarlosSilveira, Rodrigo IgnacioWe consider a natural variation of the concept of stabbing a set of segments with a simple polygon: a segment s is stabbed by a simple polygon P if at least one endpoint of s is contained in P, and a segment set S is stabbed by P if P stabs every element of S. Given a segment set S, we study the problem of finding a simple polygon P stabbing S in a way that some measure of P (such as area or perimeter) is optimized. We show that if the elements of S are pairwise disjoint, the problem can be solved in polynomial time. In particular, this solves an open problem posed by Loftier and van Kreveld [Algorithmica 56(2), 236-269 (2010)] [16] about finding a maximum perimeter convex hull for a set of imprecise points modeled as line segments. Our algorithm can also be extended to work for a more general problem, in which instead of segments, the set S consists of a collection of point sets with pairwise disjoint convex hulls. We also prove that for general segments our stabbing problem is NP-hard. (C) 2014 Elsevier B.V. All rights reserved.