Articles de revista
http://hdl.handle.net/2117/3197
20161204T04:34:19Z

The degree/diameter problem in maximal planar bipartite graphs
http://hdl.handle.net/2117/89907
The degree/diameter problem in maximal planar bipartite graphs
Dalfó Simó, Cristina; Huemer, Clemens; Salas Piñon, Julián
The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases.
20160914T10:33:44Z
Dalfó Simó, Cristina
Huemer, Clemens
Salas Piñon, Julián
The (Δ,D)(Δ,D) (degree/diameter) problem consists of finding the largest possible number of vertices nn among all the graphs with maximum degree ΔΔ and diameter DD. We consider the (Δ,D)(Δ,D) problem for maximal planar bipartite graphs, that is, simple planar graphs in which every face is a quadrangle. We obtain that for the (Δ,2)(Δ,2) problem, the number of vertices is n=Δ+2n=Δ+2; and for the (Δ,3)(Δ,3) problem, n=3Δ−1n=3Δ−1 if ΔΔ is odd and n=3Δ−2n=3Δ−2 if ΔΔ is even. Then, we prove that, for the general case of the (Δ,D)(Δ,D) problem, an upper bound on nn is approximately 3(2D+1)(Δ−2)⌊D/2⌋3(2D+1)(Δ−2)⌊D/2⌋, and another one is C(Δ−2)⌊D/2⌋C(Δ−2)⌊D/2⌋ if Δ≥DΔ≥D and CC is a sufficiently large constant. Our upper bounds improve for our kind of graphs the one given by Fellows, Hell and Seyffarth for general planar graphs. We also give a lower bound on nn for maximal planar bipartite graphs, which is approximately (Δ−2)k(Δ−2)k if D=2kD=2k, and 3(Δ−3)k3(Δ−3)k if D=2k+1D=2k+1, for ΔΔ and DD sufficiently large in both cases.

Perfect and quasiperfect domination in trees
http://hdl.handle.net/2117/86561
Perfect and quasiperfect domination in trees
Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, Maria Luz
A k quasip erfect dominating set of a connected graph G is a vertex subset S such that every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum kquasip erfect dominating set in G is denoted by 1 k ( G ) . These graph parameters were rst intro duced by Chellali et al. (2013) as a generalization of b oth the p erfect domination numb er 11 ( G ) and the domination numb er ( G ) . The study of the socalled quasip erfect domination chain 11 ( G ) 12 ( G ) 1 ( G ) = ( G ) enable us to analyze how far minimum dominating sets are from b eing p erfect. In this pap er, we provide, for any tree T and any p ositive integer k , a tight upp er b ound of 1 k ( T ) . We also prove that there are trees satisfying all p ossible equalities and inequalities in this chain. Finally a linear algorithm for computing 1 k ( T ) in any tree T is presente
20160504T10:50:08Z
Cáceres, José
Hernando Martín, María del Carmen
Mora Giné, Mercè
Pelayo Melero, Ignacio Manuel
Puertas, Maria Luz
A k quasip erfect dominating set of a connected graph G is a vertex subset S such that every vertex not in S is adjacent to at least one and at most k vertices in S . The cardinality of a minimum kquasip erfect dominating set in G is denoted by 1 k ( G ) . These graph parameters were rst intro duced by Chellali et al. (2013) as a generalization of b oth the p erfect domination numb er 11 ( G ) and the domination numb er ( G ) . The study of the socalled quasip erfect domination chain 11 ( G ) 12 ( G ) 1 ( G ) = ( G ) enable us to analyze how far minimum dominating sets are from b eing p erfect. In this pap er, we provide, for any tree T and any p ositive integer k , a tight upp er b ound of 1 k ( T ) . We also prove that there are trees satisfying all p ossible equalities and inequalities in this chain. Finally a linear algorithm for computing 1 k ( T ) in any tree T is presente

Opening the black box of energy throughputs in farm systems: a decomposition analysis between the energy returns to external inputs, internal biomass reuses and total inputs consumed (the Vallès County, Catalonia, c.1860 and 1999)
http://hdl.handle.net/2117/86061
Opening the black box of energy throughputs in farm systems: a decomposition analysis between the energy returns to external inputs, internal biomass reuses and total inputs consumed (the Vallès County, Catalonia, c.1860 and 1999)
Tello, E.; Galán, E.; Sacristán Adinolfi, Vera; Cunfer, G.; Guzmán, G. I.; González de Molina, M.; Krausmann, F.; Gringrich, S.; Padró, R.; Marco, I.; MorenoDelgado, D.
We present an energy analysis of past and present farm systems aimed to contribute to their sustainability assessment. Looking at agroecosystems as a set of energy loops between nature and society, and adopting a farmoperator standpoint at landscape level to set the system boundaries, enthalpy values of energy carriers are accounted for net Final Produce going outside as well as for Biomass Reused cycling inside, and External Inputs are accounted using embodied values. Human Labour is accounted for the fraction of the energy intake of labouring people devoted to perform farm work, considering the local or external origin of their food basket. In this approach the proportion of internal Biomass Reused becomes a hallmark of organic farm systems that tend to save External Inputs, whereas industrial farming and livestock breeding in feedlots tend to get rid of reuses replacing them with inputs coming from outside. Hence, decomposing the internal or external energy throughputs may bring to light their contrasting sociometabolic profiles. A Catalan case study in 1860 and 1990 is used as a test bench to show how revealing this decomposing analysis may be to plot the energy profiles of farm systems and their possible improvement pathways.
20160421T12:18:30Z
Tello, E.
Galán, E.
Sacristán Adinolfi, Vera
Cunfer, G.
Guzmán, G. I.
González de Molina, M.
Krausmann, F.
Gringrich, S.
Padró, R.
Marco, I.
MorenoDelgado, D.
We present an energy analysis of past and present farm systems aimed to contribute to their sustainability assessment. Looking at agroecosystems as a set of energy loops between nature and society, and adopting a farmoperator standpoint at landscape level to set the system boundaries, enthalpy values of energy carriers are accounted for net Final Produce going outside as well as for Biomass Reused cycling inside, and External Inputs are accounted using embodied values. Human Labour is accounted for the fraction of the energy intake of labouring people devoted to perform farm work, considering the local or external origin of their food basket. In this approach the proportion of internal Biomass Reused becomes a hallmark of organic farm systems that tend to save External Inputs, whereas industrial farming and livestock breeding in feedlots tend to get rid of reuses replacing them with inputs coming from outside. Hence, decomposing the internal or external energy throughputs may bring to light their contrasting sociometabolic profiles. A Catalan case study in 1860 and 1990 is used as a test bench to show how revealing this decomposing analysis may be to plot the energy profiles of farm systems and their possible improvement pathways.

Lower bounds on the maximum number of noncrossing acyclic graphs
http://hdl.handle.net/2117/86044
Lower bounds on the maximum number of noncrossing acyclic graphs
Huemer, Clemens; Mier Vinué, Anna de
This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of noncrossing spanning trees and forests. We show that the socalled double chain point configuration of N points has Omega (12.52(N)) noncrossing spanning trees and Omega (13.61(N)) noncrossing forests. This improves the previous lower bounds on the maximum number of noncrossing spanning trees and of noncrossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Toth (2013). Our analysis relies on the tools of analytic combinatorics, which enable us to count certain families of forests on points in convex position, and to estimate their average number of components. A new upper bound of O(22.12(N)) for the number of noncrossing spanning trees of the double chain is also obtained. (C) 2015 Elsevier Ltd. All rights reserved.
20160421T10:27:51Z
Huemer, Clemens
Mier Vinué, Anna de
This paper is a contribution to the problem of counting geometric graphs on point sets. More concretely, we look at the maximum numbers of noncrossing spanning trees and forests. We show that the socalled double chain point configuration of N points has Omega (12.52(N)) noncrossing spanning trees and Omega (13.61(N)) noncrossing forests. This improves the previous lower bounds on the maximum number of noncrossing spanning trees and of noncrossing forests among all sets of N points in general position given by Dumitrescu, Schulz, Sheffer and Toth (2013). Our analysis relies on the tools of analytic combinatorics, which enable us to count certain families of forests on points in convex position, and to estimate their average number of components. A new upper bound of O(22.12(N)) for the number of noncrossing spanning trees of the double chain is also obtained. (C) 2015 Elsevier Ltd. All rights reserved.

On kgons and kholes in point sets
http://hdl.handle.net/2117/85613
On kgons and kholes in point sets
Aichholzer, Oswin; Fabila Monroy, Ruy; Gonzalez Aguilar, Hernan; Hackl, Thomas; Heredia, Marco A.; Huemer, Clemens; Urrutia Galicia, Jorge; Valtr, Pavel; Vogtenhuber, Birgit
We consider a variation of the classical ErdosSzekeres problems on the existence and number of convex kgons and kholes (empty kgons) in a set of n points in the plane. Allowing the kgons to be nonconvex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any k and sufficiently large n, we give a quadratic lower bound for the number of kholes, and show that this number is maximized by sets in convex position. (C) 2014 Elsevier B.V. All rights reserved.
20160413T12:35:46Z
Aichholzer, Oswin
Fabila Monroy, Ruy
Gonzalez Aguilar, Hernan
Hackl, Thomas
Heredia, Marco A.
Huemer, Clemens
Urrutia Galicia, Jorge
Valtr, Pavel
Vogtenhuber, Birgit
We consider a variation of the classical ErdosSzekeres problems on the existence and number of convex kgons and kholes (empty kgons) in a set of n points in the plane. Allowing the kgons to be nonconvex, we show bounds and structural results on maximizing and minimizing their numbers. Most noteworthy, for any k and sufficiently large n, we give a quadratic lower bound for the number of kholes, and show that this number is maximized by sets in convex position. (C) 2014 Elsevier B.V. All rights reserved.

Blocking the kholes of point sets in the plane
http://hdl.handle.net/2117/85425
Blocking the kholes of point sets in the plane
Cano, Javier; Garcia Olaverri, Alfredo Martin; Hurtado Díaz, Fernando Alfredo; Shakai, Toshinori; Tejel Altarriba, Francisco Javier; Urrutia Galicia, Jorge
Let P be a set of n points in the plane in general position. A subset H of P consisting of k elements that are the vertices of a convex polygon is called a khole of P, if there is no element of P in the interior of its convex hull. A set B of points in the plane blocks the kholes of P if any khole of P contains at least one element of B in the interior of its convex hull. In this paper we establish upper and lower bounds on the sizes of khole blocking sets, with emphasis in the case k=5
20160408T11:58:18Z
Cano, Javier
Garcia Olaverri, Alfredo Martin
Hurtado Díaz, Fernando Alfredo
Shakai, Toshinori
Tejel Altarriba, Francisco Javier
Urrutia Galicia, Jorge
Let P be a set of n points in the plane in general position. A subset H of P consisting of k elements that are the vertices of a convex polygon is called a khole of P, if there is no element of P in the interior of its convex hull. A set B of points in the plane blocks the kholes of P if any khole of P contains at least one element of B in the interior of its convex hull. In this paper we establish upper and lower bounds on the sizes of khole blocking sets, with emphasis in the case k=5

The diameter of cyclic Kautz digraphs
http://hdl.handle.net/2117/85343
The diameter of cyclic Kautz digraphs
Böhmová, Katerina; Dalfó Simó, Cristina; Huemer, Clemens
A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs MCK(d,l) and it is derived from the Kautz digraphs K(d,l). It is wellknown that the Kautz digraphs K(d,l) have the smallest diameter among all digraphs with their number of vertices and degree. Here we define the cyclic Kautz digraphs CK(d,l), whose vertices are labeled by all possible sequences a1…al of length l , such that each character ai is chosen from an alphabet containing d+1 distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and now also requiring that a1¿al. The cyclic Kautz digraphs CK(d,l) have arcs between vertices a1a2…al and a2…alal+1, with a1¿al, a2¿al+1, and ai¿ai+1 for i=1,…,l1. The cyclic Kautz digraphs CK(d,l) are subdigraphs of the Kautz digraphs K(d,l). We give the main parameters of CK(d,l) (number of vertices, number of arcs, and diameter). Moreover, we construct the modified cyclic Kautz digraphs MCK(d,l) to obtain the same diameter as in the Kautz digraphs, and we show that MCK(d,l) are d outregular. Finally, we compute the number of vertices of the iterated line digraphs of CK(d,l).
20160407T10:39:47Z
Böhmová, Katerina
Dalfó Simó, Cristina
Huemer, Clemens
A prominent problem in Graph Theory is to find extremal graphs or digraphs with restrictions in their diameter, degree and number of vertices. Here we obtain a new family of digraphs with minimal diameter, that is, given the number of vertices and degree there is no other digraph with a smaller diameter. This new family is called modified cyclic digraphs MCK(d,l) and it is derived from the Kautz digraphs K(d,l). It is wellknown that the Kautz digraphs K(d,l) have the smallest diameter among all digraphs with their number of vertices and degree. Here we define the cyclic Kautz digraphs CK(d,l), whose vertices are labeled by all possible sequences a1…al of length l , such that each character ai is chosen from an alphabet containing d+1 distinct symbols, where the consecutive characters in the sequence are different (as in Kautz digraphs), and now also requiring that a1¿al. The cyclic Kautz digraphs CK(d,l) have arcs between vertices a1a2…al and a2…alal+1, with a1¿al, a2¿al+1, and ai¿ai+1 for i=1,…,l1. The cyclic Kautz digraphs CK(d,l) are subdigraphs of the Kautz digraphs K(d,l). We give the main parameters of CK(d,l) (number of vertices, number of arcs, and diameter). Moreover, we construct the modified cyclic Kautz digraphs MCK(d,l) to obtain the same diameter as in the Kautz digraphs, and we show that MCK(d,l) are d outregular. Finally, we compute the number of vertices of the iterated line digraphs of CK(d,l).

3colorability of pseudotriangulations
http://hdl.handle.net/2117/85279
3colorability of pseudotriangulations
Aichholzer, Oswin; Aurenhammer, Franz; Hackl, Thomas; Huemer, Clemens; Pilz, Alexander; Vogtenhuber, Birgit
Deciding 3colorability for general plane graphs is known to be an NPcomplete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudotriangulations, which are a generalization of triangulations, and prove NPcompleteness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudotriangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudotriangulations with maximum face degree four are always 3colorable. An according 3coloring can be found in linear time. Some complexity results relating to the rank of pseudotriangulations are also given.
Electronic version of an article published as International Journal of Computational Geometry & Applications, Vol. 25, No. 4 (2015) 283–298 DOI: 10.1142/S0218195915500168 © 2015 World Scientific Publishing Company. http://www.worldscientific.com/worldscinet/ijcga
20160406T10:04:57Z
Aichholzer, Oswin
Aurenhammer, Franz
Hackl, Thomas
Huemer, Clemens
Pilz, Alexander
Vogtenhuber, Birgit
Deciding 3colorability for general plane graphs is known to be an NPcomplete problem. However, for certain families of graphs, like triangulations, polynomial time algorithms exist. We consider the family of pseudotriangulations, which are a generalization of triangulations, and prove NPcompleteness for this class. This result also holds if we bound their face degree to four, or exclusively consider pointed pseudotriangulations with maximum face degree five. In contrast to these completeness results, we show that pointed pseudotriangulations with maximum face degree four are always 3colorable. An according 3coloring can be found in linear time. Some complexity results relating to the rank of pseudotriangulations are also given.

Quasiperfect domination in trees
http://hdl.handle.net/2117/85175
Quasiperfect domination in trees
Cáceres, José; Hernando Martín, María del Carmen; Mora Giné, Mercè; Pelayo Melero, Ignacio Manuel; Puertas, M. Luz
A k–quasiperfect dominating set ( k=1k=1) of a graph G is a vertex subset S such that 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 of G is denoted by ¿1k(G)¿1k(G). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept (which coincides with the case k=1k=1) and allow us to construct a decreasing chain of quasiperfect dominating parameters
¿11(G)=¿12(G)=…=¿1,¿(G)=¿(G),¿11(G)=¿12(G)=…=¿1,¿(G)=¿(G), (1)
in order to indicate how far is G from being perfectly dominated. In this work, we study general properties, tight bounds, existence and realization results involving the parameters of the socalled QPchain ( 1), for trees.
20160405T09:39:49Z
Cáceres, José
Hernando Martín, María del Carmen
Mora Giné, Mercè
Pelayo Melero, Ignacio Manuel
Puertas, M. Luz
A k–quasiperfect dominating set ( k=1k=1) of a graph G is a vertex subset S such that 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 of G is denoted by ¿1k(G)¿1k(G). Those sets were first introduced by Chellali et al. (2013) as a generalization of the perfect domination concept (which coincides with the case k=1k=1) and allow us to construct a decreasing chain of quasiperfect dominating parameters
¿11(G)=¿12(G)=…=¿1,¿(G)=¿(G),¿11(G)=¿12(G)=…=¿1,¿(G)=¿(G), (1)
in order to indicate how far is G from being perfectly dominated. In this work, we study general properties, tight bounds, existence and realization results involving the parameters of the socalled QPchain ( 1), for trees.

Completion and decomposition of hypergraphs into dominating sets of graphs
http://hdl.handle.net/2117/81940
Completion and decomposition of hypergraphs into dominating sets of graphs
Martí Farré, Jaume; Mora Giné, Mercè; Ruiz Muñoz, José Luis
The collection of the vertex dominating sets of a graph defines a hypergraph on the set of vertices of the graph. However, there are hypergraphs H that are not the collection of the vertex dominating sets of any graph. This paper deals with the question of completing these hypergraphs H to the vertex dominating sets of some graphs G. We demonstrate that such graphs G exist and, in addition, we prove that these graphs define a poset whose minimal elements provide a decomposition of H. Moreover, we show that the hypergraph H is uniquely determined by the minimal elements of this poset. The computation of such minimal elements is also discussed in some cases.
20160125T09:39:08Z
Martí Farré, Jaume
Mora Giné, Mercè
Ruiz Muñoz, José Luis
The collection of the vertex dominating sets of a graph defines a hypergraph on the set of vertices of the graph. However, there are hypergraphs H that are not the collection of the vertex dominating sets of any graph. This paper deals with the question of completing these hypergraphs H to the vertex dominating sets of some graphs G. We demonstrate that such graphs G exist and, in addition, we prove that these graphs define a poset whose minimal elements provide a decomposition of H. Moreover, we show that the hypergraph H is uniquely determined by the minimal elements of this poset. The computation of such minimal elements is also discussed in some cases.