Domingos M. Cardoso

The teacher assignment problem: A special case of the fixed charge transportation problem

A basic model of the task of assigning classes to professors, in such a way that the average number of distinct subjects assigned to each professor is minimized, is formulated as a mixed integer program. It turns out that the problem is a special case of the fixed charge transportation problem which in some cases corresponds to finding a basic solution of a transportation problem which is as degenerate as possible. We present an equivalent alternative formulation of the problem which makes it easy to prove that it is NP-hard in the strong sense and an exact branch and bound algorithm for its solution based on this alternative formulation is outlined. Computational experiments with data from a concrete problem, concludes the paper.

Laplacian eigenvectors and eigenvalues and almost equitable partitions

Relations between Laplacian eigenvectors and eigenvalues and the existence of almost equitable partitions (which are generalizations of equitable partitions) are presented. Furthermore, on the basis of some properties of the adjacency eigenvectors of a graph, a necessary and sufficient condition for the graph to be primitive strongly regular is introduced.

Spectra of graphs obtained by a generalization of the join graph operation

Abstract Taking a Fiedler’s result on the spectrum of a matrix formed from two symmetric matrices as a motivation, a more general result is deduced and applied to the determination of adjacency and Laplacian spectra of graphs obtained by a generalized join graph operation on families of graphs (regular in the case of adjacency spectra and arbitrary in the case of Laplacian spectra). Some additional consequences are explored, namely regarding the largest eigenvalue and algebraic connectivity.

Efficient edge domination in regular graphs

An induced matching of a graph G is a matching having no two edges joined by an edge. An efficient edge dominating set of G is an induced matching M such that every other edge of G is adjacent to some edge in M. We relate maximum induced matchings and efficient edge dominating sets, showing that efficient edge dominating sets are maximum induced matchings, and that maximum induced matchings on regular graphs with efficient edge dominating sets are efficient edge dominating sets. A necessary condition for the existence of efficient edge dominating sets in terms of spectra of graphs is established. We also prove that, for arbitrary fixed p>=3, deciding on the existence of efficient edge dominating sets on p-regular graphs is NP-complete.

Convex Quadratic Programming Approach to the Maximum Matching Problem

The problem of determining a maximum matching or whether there exists a perfect matching, is very common in a large variety of applications and as been extensively studied in graph theory. In this paper we start to introduce a characterisation of a family of graphs for which its stability number is determined by convex quadratic programming. The main results connected with the recognition of this family of graphs are also introduced. It follows a necessary and sufficient condition which characterise a graph with a perfect matching and an algorithmic strategy, based on the determination of the stability number of line graphs, by convex quadratic programming, applied to the determination of a perfect matching. A numerical example for the recognition of graphs with a perfect matching is described. Finally, the above algorithmic strategy is extended to the determination of a maximum matching of an arbitrary graph and some related results are presented.

Dominating Induced Matchings

We study the problem of determining whether or not a graph G has an induced matching that dominates every edge of the graph, which is also known as efficient edge domination . This problem is known to be NP-complete in general as well as in some restricted domains, such as bipartite graphs or regular graphs. In this paper, we identify a graph parameter to which the complexity of the problem is sensible and produce results of both negative (intractable) and positive (solvable in polynomial time) type.

A generalization of the Hoffman - Lovász upper boundon the independence number of a regular graph

A family of quadratic programming problems whose optimal values are upper boundson the independence number of a graph is introduced. Among this family, the quadraticprogramming problem which gives the best upper bound is identified. Also the proof thatthe upper bound introduced by Hoffman and Lovász for regular graphs is a particular caseof this family is given. In addition, some new results characterizing the class of graphs forwhich the independence number attains the optimal value of the above best upper bound aregiven. Finally a polynomial-time algorithm for approximating the size of the maximumindependent set of an arbitrary graph is described and the computational experiments carriedout on 36 DIMACS clique benchmark instances are reported.

A generalization of Fiedler's lemma and some applications

In a previous paper [M. Robbiano, E.A. Martins, and I. Gutman, Extending a theorem by Fiedler and applications to graph energy, MATCH Commun. Math. Comput. Chem. 64 (2010), pp. 145–156], a lemma by Fiedler was used to obtain eigenspaces of graphs, and applied to graph energy. In this article Fiedlers lemma is generalized and this generalization is applied to graph spectra and graph energy.

The generalized simplex method

A method for solving linear programs which corresponds to a generalization of the simplex algorithm is introduced. This method makes feasible movements between faces of arbitrary dimension of a polytope and converges to an optimal face. The approach starts at a face of a high dimension, which is easy to determine, and systematically moves from one face to another such that, in general, the dimension of the current face decreases in each iteration.

Spectral upper bounds on the size of k-regular induced subgraphs

Abstract Convex quadratic programming upper bounds on the size of k -regular induced subgraphs are analyzed and a necessary and sufficient condition for such upper bounds being tight is introduced. Based on this approach, new spectral upper bounds on the order of maximum size k -regular induced subgraphs are deduced. Related open problems and a few computational experiments are presented.