No abstract

Given a connected graph G, the metric (resp. edge metric) dimension of G is the cardinality of the smallest ordered set of vertices that uniquely identifies every pair of distinct vertices (resp. edges) of G by means of distance vectors to such a set. In this work, we settle three open problems on (edge) metric dimension of graphs. Specifically, we show that for every r, t ≥ 2 with r = t, there is n 0 , such that for every n ≥ n 0 there exists a graph G of order n with metric dimension r and edge metric dimension t, which among other consequences, shows the existence of infinitely many graph whose edge metric dimension is strictly smaller than its metric dimension. In addition, we also prove that it is not possible to bound the edge metric dimension of a graph G by some constant factor of the metric dimension of G.

The metric (resp. edge metric or mixed metric) dimension of a graph G, is the cardinality of the smallest ordered set of vertices that uniquely recognizes all the pairs of distinct vertices (resp. edges, or vertices and edges) of G by using a vector of distances to this set. In this note we show two unexpected results on hypercube graphs. First, we show that the metric and edge metric dimension of Q d differ by only one for every integer d. In particular, if d is odd, then the metric and edge metric dimensions of Q d are equal. Second, we prove that the metric and mixed metric dimensions of the hypercube Q d are equal for every d ≥ 3. We conclude the paper by conjecturing that all these three types of metric dimensions of Q d are equal when d is large enough.

In this note we show two unexpected results concerning the metric, the edge metric and the mixed metric dimensions of hypercube graphs. First, we show that the metric and the edge metric dimensions of Q d differ by at most one for every integer d. In particular, if d is odd, then the metric and the edge metric dimensions of Q d are equal. Second, we prove that the metric and the mixed metric dimensions of the hypercube Q d are equal for every d ≥ 3. We conclude the paper by conjecturing that all these three types of metric dimensions of Q d are equal when d is large enough.