Guillermo Durán

NP-completeness results for edge modification problems

The aim of edge modification problems is to change the edge set of a given graph as little as possible in order to satisfy a certain property. Edge modification problems in graphs have a lot of applications in different areas, and many polynomial-time algorithms and NP-completeness proofs for this kind of problems are known. In this work we prove new NP-completeness results for these problems in some graph classes, such as interval, circular-arc, permutation, circle, bridged, weakly chordal and clique-Helly graphs.

On Balanced Graphs

Berge defined a hypergraph to be balanced if its incidence matrix is balanced. We consider this concept applied to graphs, and call a graph to be balanced when its clique matrix is balanced. Characterizations of balanced graphs by forbidden subgraphs and by clique subgraphs are proved in this work. Using properties of domination we define four subclasses of balanced graphs. Two of them are characterized by 0–1 matrices and can be recognized in polynomial time. Furthermore, we propose polynomial time combinatorial algorithms for the problems of stable set, clique-independent set and clique-transversal for one of these subclasses of balanced graphs. Finally, we analyse the behavior of balanced graphs and these four subclasses under the clique graph operator.

On the b-Coloring of Cographs and P 4 -Sparse Graphs

A b-coloring of a graph is a coloring such that every color class admits a vertex adjacent to at least one vertex receiving each of the colors not assigned to it. The b-chromatic number of a graph G, denoted by χb(G), is the maximum number t such that G admits a b-coloring with t colors. A graph G is b-continuous if it admits a b-coloring with t colors, for every

Polynomial time recognition of unit circular-arc graphs

t = \chi(G), \ldots, \chi_b(G)

Partial characterizations of clique-perfect graphs I: Subclasses of claw-free graphs

. We define a graph G to be b-monotonic if χb(H1) ≥ χb(H2) for every induced subgraph H1 of G, and every induced subgraph H2 of H1. In this work, we prove that P4-sparse graphs (and, in particular, cographs) are b-continuous and b-monotonic. Besides, we describe a dynamic programming algorithm to compute the b-chromatic number in polynomial time within these graph classes.

Partial characterizations of clique-perfect graphs II: Diamond-free and Helly circular-arc graphs

A clique-transversal of a graph G is a subset of vertices intersecting all the cliques of G. A clique-independent set is a subset of pairwise disjoint cliques of G. Denote by τC(G) and αC(G) the cardinalities of the minimum clique-transversal and maximum clique-independent set of G, respectively. Say that G is clique-perfect when τC(H)=αC(H), for every induced subgraph H of G. In this paper, we prove that every graph not containing a 4-wheel nor a 3-fan as induced subgraphs and such that every odd cycle of length greater than 3 has a short chord is clique-perfect. The proof leads to polynomial time algorithms for finding the parameters τC(G) and αC(G), for graphs belonging to this class. In addition, we prove that to decide whether or not a given subset of vertices of a graph is a clique-transversal is Co-NP-Complete. The complexity of this problem has been mentioned as unknown in the literature. Finally, we describe a family of highly clique-imperfect graphs, that is, a family of graphs G whose difference τC(G)−αC(G) is arbitrarily large.

A branch-and-cut algorithm for scheduling the highly-constrained Chilean soccer tournament

We present an efficient algorithm for recognizing unit circular-arc (UCA) graphs, based on a characterization theorem for UCA graphs proved by Tucker in the seventies. Given a proper circular-arc (PCA) graph G, the algorithm starts from a PCA model for G, removes all its circle-covering pairs of arcs and determines whether G is a UCA graph. We also give an O(N) time bound for Tuckers 3/2-approximation algorithm for coloring circular-arc graphs with N vertices, when a circular-arc model is given.

An approach for efficient ship routing

A method is proposed that uses operations research techniques to optimize the routes of waste collection vehicles servicing dumpster or skip-type containers. The waste collection problem is reduced to the classic travelling salesman problem, which is then solved using the Concorde solver program. A case study applying the method to the collection system in the southern zone of Buenos Aires is also presented. In addition to the typical minimum distance criterion, the optimization problem incorporates the objective of reducing vehicle wear and tear as measured by the physics concept of mechanical work. The solution approach, employing graph theory and mathematical programming tools, is fully described and the data correction process is also discussed. The application of the proposed method minimized the distance travelled by each collection vehicle in the areas studied, with actual reductions ranging from 10 to 40% of the existing routes. The shortened distances led in turn to substantial decreases in work done and therefore in vehicle wear and tear. Extrapolation of the results to the entire southern zone of Buenos Aires indicates potential savings for the civic authorities of more than US