On the metric dimension of some families of graphs

The concept of (minimum) resolving set has proved to be useful and/or related to a variety of fields such as Chemistry [G. Chartrand, D. Erwin, G. L. Johns and P. Zhang, Boundary vertices in graphs, Discrete Math. 263 (2003) 25-34; C. Poisson and P. Zhang, The metric dimension of unicyclic graphs, J. Comb. Math Comb. Comput. 40 (2002) 17–32], Robotic Navigation [S. Khuller, B. Raghavachari and A. Rosenfeld, Landmarks in graphs, Disc. Appl. Math. 70 (1996) 217–229; B. Shanmukha, B. Sooryanarayana and K. S. Harinath, Metric dimension of wheels, Far East J. Appl. Math. 8 (3) (2002) 217–229] and Combinatorial Search and Optimization [A. Sebö and E. Tannier, On metric generators of graphs, Math. Oper. Res. 29 (2) (2004) 383–393]. This work is devoted to evaluating the so-called metric dimension of a finite connected graph, i.e., the minimum cardinality of a resolving set, for a number of graph families, as long as to study its behavior with respect to the join and the cartesian product of graphs.