For a large class of metric spaces X including discrete groups we prove that the asymptotic Assouad-Nagata dimension AN-asdim X of X coincides with the covering dimension dim(ν L X) of the Higson corona of X with respect to the sublinear coarse structure on X. Then we apply this fact to prove the equality AN-asdim(X × R) = AN-asdim X + 1. We note that the similar equality for Gromov's asymptoti...

A resolving set of a graph G is a set S ⊆ V (G), such that, every pair of distinct vertices of G is resolved by some vertex in S. The metric dimension of G, denoted by β(G), is the minimum cardinality of all the resolving sets of G. Shamir Khuller et al. [10], in 1996, proved that a graph G with β(G) = 2 can have neither K5 nor K3,3 as its subgraph. In this paper, we obtain a forbidden subgraph...