127 research outputs found
The Simultaneous Strong Resolving Graph and the Simultaneous Strong Metric Dimension of Graph Families
We consider in this work a new approach to study the simultaneous strong metric dimension of graphs families, while introducing the simultaneous version of the strong resolving graph. In concordance, we consider here connected graphs G whose vertex sets are represented as V(G), and the following terminology. Two vertices u,v is an element of V(G) are strongly resolved by a vertex w is an element of V(G), if there is a shortest w-v path containing u or a shortest w-u containing v. A set A of vertices of the graph G is said to be a strong metric generator for G if every two vertices of G are strongly resolved by some vertex of A. The smallest possible cardinality of any strong metric generator (SSMG) for the graph G is taken as the strong metric dimension of the graph G. Given a family F of graphs defined over a common vertex set V, a set S subset of V is an SSMG for F, if such set S is a strong metric generator for every graph G is an element of F. The simultaneous strong metric dimension of F is the minimum cardinality of any strong metric generator for F, and is denoted by Sds(F). The notion of simultaneous strong resolving graph of a graph family F is introduced in this work, and its usefulness in the study of Sds(F) is described. That is, it is proved that computing Sds(F) is equivalent to computing the vertex cover number of the simultaneous strong resolving graph of F. Several consequences (computational and combinatorial) of such relationship are then deduced. Among them, we remark for instance that we have proved the NP-hardness of computing the simultaneous strong metric dimension of families of paths, which is an improvement (with respect to the increasing difficulty of the problem) on the results known from the literature
The simultaneous (strong) metric dimension of graph families
En aquesta tesi vam introduir la noció de resolubilitat simultà nia per a famÃlies de grafs definides sobre un conjunt de vèrtexs en comú. Els principals resultats de la tesi s'han abordat als generadors i bases mètrics simultanis, aixà com la dimensió mètrica simultà nia d'aquestes famÃlies. Addicionalment, hem tractat dues formes de resolubilitat simultà nia relacionades. Primerament, vam abordar la dimensió d'adjacència simultà nia, la qual va demostrar la seva utilitat per caracteritzar la dimensió mètrica simultà nia de famÃlies compostes per grafs-producte lexicogrà fics i corona. En segon lloc, vam estudiar les propietats principals de la dimensió mètrica forta simultà nia. En tots els casos, el focus va estar a determinar les cotes generals per a aquests parà metres, les seves relacions amb els parà metres de resolubilitat està ndard dels grafs individuals i, quan va ser possible, donar valors exactes o cotes ajustades per certes famÃlies especÃfiques.
Des del punt de vista computacional, el problema encara no es pot considerar resolt per al cas general, ja que vam aconseguir verificar que el requisit de simultaneïtat augmenta la complexitat computacional dels cà lculs relacionats amb aquests parà metres, els quals ja s'havia demostrat que eren NP -difÃcils. En particular, vam caracteritzar famÃlies compostes per grafs pels quals alguns parà metres està ndards de resolubilitat es poden calcular eficientment, mentre que calcular els parà metres simultanis associats és NP-difÃcil. Per pal•liar aquest problema, vam proposar diversos mètodes per estimar aproximadament aquests parà metres i vam realitzar una avaluació experimental per estudiar el seu comportament en col•leccions de famÃlies de grafs generades aleatòriament.En esta tesis hemos introducido la noción de resolubilidad simultánea para familias de grafos definidas sobre un conjunto de vértices en común. Los principales resultados de la tesis han abordado los generadores y bases métricos simultáneos, asà como la dimensión métrica simultánea de dichas familias. Adicionalmente, hemos tratado dos formas de resolubilidad simultánea relacionadas. Primeramente, abordamos la dimensión de adyacencia simultánea, la cual demostró su utilidad para caracterizar la dimensión métrica simultánea de familias compuestas por grafos-producto lexicográficos y corona. En segundo lugar, estudiamos las propiedades principales de la dimensión métrica fuerte simultánea. En todos los casos, el foco estuvo en determinar las cotas generales para estos parámetros, sus relaciones con los parámetros de resolubilidad estándar de los grafos individuales y, cuando fue posible, dar valores exactos o cotas ajustadas para ciertas familias especÃficas.
Desde el punto de vista computacional, los problemas aún no se pueden considerar resueltos para el caso general, ya que logramos verificar que el requisito de simultaneidad aumenta la complejidad computacional de los cálculos relacionados con estos parámetros, los cuales ya se habÃa demostrado que eran NP-difÃciles. En particular, caracterizamos familias compuestas por grafos para los cuales algunos parámetros estándares de resolubilidad se pueden calcular eficientemente, mientras que calcular los parámetros simultáneos asociados es NP-difÃcil. Para paliar este problema, propusimos varios métodos para estimar aproximadamente estos parámetros y realizamos una evaluación experimental para estudiar su comportamiento en colecciones de familias de grafos generadas aleatoriamente.In this thesis we have introduced the notion of simultaneous resolvability for graph families defined on a common vertex set. The main results of the thesis have dealt with simultaneous metric generators and bases, as well as the simultaneous metric dimension of such families. Additionally, we have covered two related forms of simultaneous resolvability. Firstly, we treated the simultaneous adjacency dimension, which proved useful for characterizing the simultaneous metric dimension of families composed by lexicographic and corona product graphs. Secondly, we studied the main properties of the simultaneous strong metric dimension. In all cases, our focus was on determining the general bounds for these parameters, their relations to the standard resolvability parameters of the individual graphs and, when possible, giving exact values or sharp bounds for a number of specific families.
Computationally, these problems are far from solved for the general case, as we were able to verify that the requirement of simultaneity adds on the complexity of the calculations involving these resolvability parameters, which had already been proven to be NP-hard for their standard counterparts. In particular, we characterized families composed by graphs for which some standard resolvability parameters can be efficiently computed, while computing the associated simultaneous parameters is NP-hard. To alleviate this problem, we proposed several methods for approximately estimating these parameters and conducted an experimental evaluation to study their behaviour on randomly generated collections of graph families
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