On the local metric dimension of graphs

Identificador:  TDX:2573

Handle: https://hdl.handle.net/20.500.11797/TDX2573

Autors:  Barragán Ramírez, Gabriel Antonio

Resum:
The metric dimension of a general metric space was introduced in 1953 but attracted little attention until, about twenty years later, it was applied to the distances between vertices of a graph. Since then it has been frequently used in graph theory, chemistry, biology, robotics and many other disciplines. Due to the variety of situations from which the problem of distinguishing the vertices of a graph can arise, several variants of the original concept of metric dimension have been appearing in specialized literature. In this thesis we study one of these variants, namely, the local metric dimension. Specifically, we focus on the problem of computing the local metric dimension of graphs. We first report on the state of the art on the local metric dimension and present some original results in which we characterize all graphs that reach some known bounds. Secondly, we obtain closed formulas and tight bounds on the local metric dimension of several families of graphs, including strong product graphs, corona product graphs, rooted product graphs and lexicographic product graphs. Finally, we introduce the study of simultaneous local metric dimension and we give some general results on this new research line.