On upper dimension of graphs and their bases sets

The metric representation of a vertex v with respect to an ordered subset W = {w(1), w(2),..., w(n)} subset of V (G) is an ordered k-tuple defined by r (v vertical bar W) = (d(v, w(1)), d(v, w(2)),..., d (v, w(n))), where d(u, v) denotes the distance between the vertices a and v. A subset W subset of V (G) is a resolving set if all vertices of G have distinct representations with respect to W. A resolving set of the largest order whose no proper subset resolves all vertices of G is called the upper basis of G and the cardinality of the upper basis is called the upper dimension of G. A vertex v having at least one pendent edge incident on it is called a star vertex and the number of pendent edges incident on a vertex v is called the star degree of v. We determine the upper dimension of certain families of graphs and characterize the cases in which upper dimension equals the metric dimension. For instance, it is shown that metric dimension equals upper dimension for the graphs defined by the Cartesian product of K-n and K-2 and for trees having no star vertices of star degree 1. Further, it is also shown that the upper dimension of a graph equals its metric dimension if the vertex set of G can be partitioned into distance similar equivalence classes.