Contractors for flows

We answer a question raised by Lov\'asz and B. Szegedy [Contractors and connectors in graph algebras, J. Graph Theory 60:1 (2009)] asking for a contractor for the graph parameter counting the number of B-flows of a graph, where B is a subset of a finite Abelian group closed under inverses. We prove our main result using the duality between flows and tensions and finite Fourier analysis. We exhibit several examples of contractors for B-flows, which are of interest in relation to the family of B-flow conjectures formulated by Tutte, Fulkerson, Jaeger, and others.Comment: 22 pages, 1 figur

The resolving number of a graph

We study a graph parameter related to resolving sets and metric dimension, namely the resolving number, introduced by Chartrand, Poisson and Zhang. First, we establish an important difference between the two parameters: while computing the metric dimension of an arbitrary graph is known to be NP-hard, we show that the resolving number can be computed in polynomial time. We then relate the resolving number to classical graph parameters: diameter, girth, clique number, order and maximum degree. With these relations in hand, we characterize the graphs with resolving number 3 extending other studies that provide characterizations for smaller resolving number.Comment: 13 pages, 3 figure

Resolving sets for breaking symmetries of graphs

This paper deals with the maximum value of the difference between the determining number and the metric dimension of a graph as a function of its order. Our technique requires to use locating-dominating sets, and perform an independent study on other functions related to these sets. Thus, we obtain lower and upper bounds on all these functions by means of very diverse tools. Among them are some adequate constructions of graphs, a variant of a classical result in graph domination and a polynomial time algorithm that produces both distinguishing sets and determining sets. Further, we consider specific families of graphs where the restrictions of these functions can be computed. To this end, we utilize two well-known objects in graph theory: $k$-dominating sets and matchings.Comment: 24 pages, 12 figure

Graph homomorphisms, the Tutte polynomial and “q-state Potts uniqueness”

We establish for which weighted graphs H homomorphism functions from multigraphs G to H are specializations of the Tutte polynomial of G, answering a question of Freedman, Lov´asz and Schrijver. We introduce a new property of graphs called “q-state Potts uniqueness” and relate it to chromatic and Tutte uniqueness, and also to “chromatic–flow uniqueness”, recently studied by Duan, Wu and Yu.Ministerio de Educación y Ciencia MTM2005-08441-C02-0

Distinguishing graphs by their left and right homomorphism profiles

We introduce a new property of graphs called ‘q-state Potts unique-ness’ and relate it to chromatic and Tutte uniqueness, and also to ‘chromatic–flow uniqueness’, recently studied by Duan, Wu and Yu. We establish for which edge-weighted graphs H homomor-phism functions from multigraphs G to H are specializations of the Tutte polynomial of G, in particular answering a question of Freed-man, Lovász and Schrijver. We also determine for which edge-weighted graphs H homomorphism functions from multigraphs G to H are specializations of the ‘edge elimination polynomial’ of Averbouch, Godlin and Makowsky and the ‘induced subgraph poly-nomial’ of Tittmann, Averbouch and Makowsky. Unifying the study of these and related problems is the notion of the left and right homomorphism profiles of a graph.Ministerio de Educación y Ciencia MTM2008-05866-C03-01Junta de Andalucía FQM- 0164Junta de Andalucía P06-FQM-0164

Computing Optimal Shortcuts for Networks

We study augmenting a plane Euclidean network with a segment, called shortcut, to minimize the largest distance between any two points along the edges of the resulting network. Questions of this type have received considerable attention recently, mostly for discrete variants of the problem. We study a fully continuous setting, where all points on the network and the inserted segment must be taken into account. We present the first results on the computation of optimal shortcuts for general networks in this model, together with several results for networks that are paths, restricted to two types of shortcuts: shortcuts with a fixed orientation and simple shortcuts

Enseñanza basada en contenidos: una experiencia para el desarrollo de competencias del EEES en grados en Ingeniería Informática

En esta ponencia describimos una estrategia docente diseñada para potenciar la implicación del alumno en el desarrollo de las clases. Aprovechando la creación de grupos con docencia en inglés en los grados en Ingeniería Informática, hemos utilizado en nuestra asignatura una metodología activa basada en la participación del alumno durante las clases. En estos grupos, un objetivo adicional es que el alumno progrese en su dominio del inglés, que claramente se alcanzará si el alumno practica activamente el idioma. Siguiendo esta idea, hemos enfocado nuestra asignatura de forma que el alumno sea el protagonista de las clases, como se indica en el Espacio Europeo de Educación Superior (EEES), realizando diversas actividades durante el curso. Con este enfoque hemos obtenido buenos resultados finales, a nivel de calificaciones y de dominio del idioma, además de desarrollar otras competencias del EEES (capacidad de exposición en público, trabajo en grupo, actitud crítica, etc).SUMMARY -- This paper describes a teaching approach designed to promote the involvement of students in class. Due to the creation of English-teaching groups in Computer Engineering Degrees, we have used an active methodology based on the participation of students in class. In these groups, an additional goal is the progress of students in their fluency in English, an aspect that will be clearly achieved if they actively practice this language. Following this idea, we have oriented our subject so that the student plays an active role in class, as indicated in the European Higher Education Area (EHEA), by means of different activities during the semester. With this approach, we have obtained good final results in terms of qualifications and English level, and other skills indicated by the EHEA (such as public presentation skills, cooperative work, critical attitude,...).Peer Reviewe

Continuous mean distance of a weighted graph

We study the concept of the continuous mean distance of a weighted graph. For connected unweighted graphs, the mean distance can be defined as the arithmetic mean of the distances between all pairs of vertices. This parameter provides a natural measure of the compactness of the graph, and has been intensively studied, together with several variants, including its version for weighted graphs. The continuous analog of the (discrete) mean distance is the mean of the distances between all pairs of points on the edges of the graph. Despite being a very natural generalization, to the best of our knowledge this concept has been barely studied, since the jump from discrete to continuous implies having to deal with an infinite number of distances, something that increases the difficulty of the parameter. In this paper we show that the continuous mean distance of a weighted graph can be computed in time quadratic in the number of edges, by two different methods that apply fundamental concepts in discrete algorithms and computational geometry. We also present structural results that allow a faster computation of this continuous parameter for several classes of weighted graphs. Finally, we study the relation between the (discrete) mean distance and its continuous counterpart, mainly focusing on the relevant question of the convergence when iteratively subdividing the edges of the weighted graph