Grundy dominating sequences on X-join product

In this paper we study the Grundy domination number on the X-join product G↩R of a graph G and a family of graphs R={Gv:v∈V(G)}. The results led us to extend the few known families of graphs where this parameter can be efficiently computed. We prove that if, for all v∈V(G), the Grundy domination number of Gv is given, and G is a power of a cycle, a power of a path, or a split graph, computing the Grundy domination number of G↩R can be done in polynomial time. In particular, our results for powers of cycles and paths are derived from a polynomial reduction to the Maximum Weight Independent Set problem on these graphs. As a consequence, we derive closed formulas to compute the Grundy domination number of the lexicographic product G∘H when G is a power of a cycle, a power of a path or a split graph, generalizing the results on cycles and paths given by Brešar et al. in 2016. Moreover, our results on the X-join product when G is a split graph also provide polynomial-time algorithms to compute the Grundy domination number for (q,q−4) graphs, partner limited graphs and extended P4-laden graphs, graph classes that are high in the hierarchy of few P4’s graphs.Fil: Nasini, Graciela Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Torres, Pablo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin

A DSATUR-based algorithm for the Equitable Coloring Problem

This paper describes a new exact algorithm for the Equitable Coloring Problem, a coloring problem where the sizes of two arbitrary color classes differ in at most one unit. Based on the well known DSatur algorithm for the classic Coloring Problem, a pruning criterion arising from equity constraints is proposed and analyzed. The good performance of the algorithm is shown through computational experiments over random and benchmark instances.Fil: Méndez-Díaz, Isabel. Universidad de Buenos Aires. Facultad de Ciencias Exactas y Naturales. Departamento de Computación; ArgentinaFil: Nasini, Graciela Leonor. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Severin, Daniel Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentin

A Branch and Price Algorithm for List Coloring Problem

Coloring problems in graphs have been used to model a wide range of real applications. In particular, the List Coloring Problem generalizes the well-known Graph Coloring Problem for which many exact algorithms have been developed. In this work, we present a Branch-and-Price algorithm for the weighted version of the List Coloring Problem, based on the one developed by Mehrotra and Trick (1996) for the Graph Coloring Problem. This version considers non-negative weights associated to each color and it is required to assign a color to each vertex from predetermined lists in such a way the sum of weights of the assigned colors is minimum. Computational experiments show the good performance of our approach, being able to comfortably solve instances whose graphs have up to seventy vertices. These experiences also bring out that the hardness of the instances of the List Coloring Problem does not seem to depend only on quantitative parameters such as the size of the graph, its density, and the size of list of colors, but also on the distribution of colors present in the lists.Fil: Lucci, Mauro. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Nasini, Graciela Leonor. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Severin, Daniel Esteban. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentina10th Latin and American Algorithms, Graphs and Optimization Symposium (LAGOS 2019)Belo HorizonteBrasilCoordenação de Aperfeiçoamento de Pessoal de Nivel SuperiorConselho Nacional de Desenvolvimento Científico e Técnologico do BrasilUniversidade Federal de Minas Gerai

Planning the workday of bus drivers by a graph list-coloring model

In this work, we address the problem of planning the workday of bus drivers in argentinian intercity bus transport companies. In particular, we focus on a company which needs to fulfill roughly 800 trips per day between 3 cities of the Province of Buenos Aires with a stuff of around 200 drivers and 100 buses. Planning consists of assigning one driver to each trip in a way the driver performs all the trips without scheduling conflicts and minimizing the overall amount of overtime among all bus drivers. We model the problem as a particular Graph Coloring Problem and we propose an Integer Linear Programming formulation. Computations experiments show that this formulation outperforms other ones given in the literature for the same problem. In order to address large instances as the one given by the company, we also propose a heuristic algorithm that delivers better solutions than the company actually uses in a reasonably amount of time. The heuristic has two phases where the first one constructs an initial solution and the second one improves the solution iteratively.Fil: Lucci, Mauro. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Nasini, Graciela Leonor. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Severin, Daniel Esteban. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentin

Total dominating sequences in trees, split graphs, and under modular decomposition

A sequence of vertices in a graph G with no isolated vertices is called a total dominating sequence if every vertex in the sequence totally dominates at least one vertex that was not totally dominated by preceding vertices in the sequence, and, at the end all vertices of G are totally dominated (by definition a vertex totally dominates its neighbors). The maximum length of a total dominating sequence is called the Grundy total domination number, γgr t(G), of G, as introduced in Brešar et al. (2016). In this paper we continue the investigation of this concept, mainly from the algorithmic point of view. While it was known that the decision version of the problem is NP-complete in bipartite graphs, we show that this is also true if we restrict to split graphs. A linear time algorithm for determining the Grundy total domination number of an arbitrary forest T is presented, based on the formula γgr t(T)=2τ(T), where τ(T) is the vertex cover number of T. A similar efficient algorithm is presented for bipartite distance-hereditary graphs. Using the modular decomposition of a graph, we present a frame for obtaining polynomial algorithms for this problem in classes of graphs having relatively simple modular subgraphs. In particular, a linear algorithm for determining the Grundy total domination number of P4-tidy graphs is presented. In addition, we prove a realization result by exhibiting a family of graphs Gk such that γgr t(Gk)=k, for any k∈Z+∖{1,3}, and showing that there are no graphs G with γgr t(G)∈{1,3}. We also present such a family, which has minimum possible order and size among all graphs with Grundy total domination number equal to k.Fil: Brešar, Boštjan. University of Maribor; Eslovenia. Institute of Mathematics, Physics and Mechanics; EsloveniaFil: Kos, Tim. Institute of Mathematics, Physics and Mechanics; EsloveniaFil: Nasini, Graciela Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Torres, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin

Circuits and Circulant Minors

Circulant contraction minors play a key role for characterizing ideal circular matrices in terms of minimally non ideal structures. In this article we prove necessary and sufficient conditions for a circular matrix A to have circulant contraction minors in terms of circuits in a digraph associated with A. In the particular case when A itself is a circulant matrix, our result provides an alternative characterization to the one previously known from the literatureFil: Bianchi, Silvia. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Nasini, Graciela Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Tolomei, Paola Beatriz. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Torres, Luis Miguel. Escuela Politécnica Nacional; Ecuador10th Latin and American Algorithms, Graphs and Optimization SymposiumBelo HorizonteBrasilCoordenação de Aperfeiçoamento de Pessoal de Nivel SuperiorConselho Nacional de Desenvolvimento Científico e Técnologico do BrasilUniversidade Federal de Minas Gerai

On dominating set polyhedra of circular interval graphs

Clique-node and closed neighborhood matrices of circular interval graphs are circular matrices. The stable set polytope and the dominating set polytope on these graphs are therefore closely related to the set packing polytope and the set covering polyhedron on circular matrices. Eisenbrand et al. [18] take advantage of this relationship to propose a complete linear description of the stable set polytope on circular interval graphs. In this paper we follow similar ideas to obtain a complete description of the dominating set polytope on the same class of graphs. As in the packing case, our results are established for a larger class of covering polyhedra of the form Q ∗ (A, b) := conv {x ∈ Z n + : Ax ≥ b}, with A a circular matrix and b an integer vector. These results also provide linear descriptions of polyhedra associated with several variants of the dominating set problem on circular interval graphs.Fil: Bianchi, Silvia María. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; ArgentinaFil: Nasini, Graciela Leonor. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Tolomei, Paola Beatriz. Universidad Nacional de Rosario. Facultad de Ciencias Exactas, Ingeniería y Agrimensura; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; ArgentinaFil: Torres, Luis Miguel. Escuela Politécnica Nacional; Ecuado

Limited Packing and Multiple Domination problems: Polynomial time reductions

The Limited Packing and Multiple Domination problems in graphs have closely-related definitions and the same computational complexity on several graph classes. In this work we present two polynomial time reductions between them. Besides, we take into consideration generalized versions of these problems and obtain polynomial time reductions between each one and its generalized version.Fil: Leoni, Valeria Alejandra. Universidad Nacional de Rosario; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; ArgentinaFil: Nasini, Graciela Leonor. Universidad Nacional de Rosario; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentin

Some links between identifying codes and separating, dominating and total dominating sets in graphs

In the search for a dynamic programming-based algorithm derived from the modular decomposition of graphs, we analyze the behavior of the identifying code number under disjoint union and join operations. This study lead us to investigate the behavior of new parameters related to separating, dominating and total dominating sets under the same operations. The obtained results and the modular decomposition of graphs easily result in a dynamic programming-based algorithm to calculate the identifying code number (and the related parameters) of a graph from the parameter values of its modular subgraphs. In particular, we obtain closed formulas for the parameters on spider and quasi-spider graphs which allow us to derive a simple and easy-to-implement linear time algorithm to obtain the identifying code number (and the related parameters) of P4-tidy graphs.Fil: Nasini, Graciela Leonor. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; ArgentinaFil: Torres, Pablo Daniel. Consejo Nacional de Investigaciones Científicas y Técnicas; Argentina. Universidad Nacional de Rosario. Facultad de Ciencias Exactas Ingeniería y Agrimensura. Escuela de Ciencias Exactas y Naturales. Departamento de Matemática; Argentin

Combinatorial Flexibility problems and their computational Complexity

The concept of flexibility-originated in the context of heat exchanger networks-is associated with a substructure which guarantees the performance of the original structure, in a given range of possible states. We extend this concept to combinatorial optimization problems, and prove several computational complexity results in this new framework. Under some monotonicity conditions, we prove that a combinatorial optimization problem polynomially transforms to its associated flexibility problem, but that the converse need not be true. In order to obtain polynomial flexibility problems, we have to restrict ourselves to combinatorial optimization problems on matroids. We also prove that, when relaxing in different ways the matroid structure, the flexibility problems become NP-complete. This fact is shown by proving the NP-completeness of the flexibility problems associated with the Shortest Path, Minimum Cut and Weighted Matching problems.Fil: Aguilera, Néstor Edgardo. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Santa Fe. Instituto de Matemática Aplicada del Litoral. Universidad Nacional del Litoral. Instituto de Matemática Aplicada del Litoral; ArgentinaFil: Leoni, Valeria Alejandra. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentina. Universidad Nacional de Rosario; ArgentinaFil: Nasini, Graciela Leonor. Universidad Nacional de Rosario; Argentina. Consejo Nacional de Investigaciones Científicas y Técnicas. Centro Científico Tecnológico Conicet - Rosario; Argentin