We compute the asymptotic dimension of the rationals given with an invariant proper metric. Also we show that a countable torsion abelian group taken with an invariant proper metric has asymptotic dimension zero.

We introduce the concept of effective dimension for a general metric space. Effective dimension was defined by Lutz in (Lutz 2003) for Cantor space and has also been extended to Euclidean space. Our extension to other metric spaces is based on a supergale characterization of Hausdorff dimension. We present here the concept of constructive dimension and its characterization in terms of Kolmogoro...

We study the approximation complexity of the Metric Dimension problem in bounded degree, dense as well as in general graphs. For the general case, we prove that the Metric Dimension problem is not approximable within (1 − ǫ) lnn for any ǫ > 0, unless NP ⊆ DTIME(nlog logn), and we give an approximation algorithm which matches the lower bound. Even for bounded degree instances it is APX-hard to d...

A resolving set for a graph Γ is a collection of vertices S, chosen so that for each vertex v, the list of distances from v to the members of S uniquely specifies v. The metric dimension of Γ is the smallest size of a resolving set for Γ. Much attention has been paid to the metric dimension of distance-regular graphs. Work of Babai from the early 1980s yields general bounds on the metric dimens...

A set $Wsubset V (G)$ is called a resolving set, if for every two distinct vertices $u, v in V (G)$ there exists $win W$ such that $d(u,w) not = d(v,w)$, where $d(x, y)$ is the distance between the vertices $x$ and $y$. A resolving set for $G$ with minimum cardinality is called a metric basis. A graph with a unique metric basis is called a uniquely dimensional graph. In this paper, we establish...

For an undirected graph G = (V,E), a vertex τ ∈ V separates vertices u and v (where u, v ∈ V , u 6= v) if their distances to τ are not equal. Given an integer parameter k ≥ 1, a set of vertices L ⊆ V is a feasible solution if for every pair of distinct vertices, u, v, there are at least k distinct vertices τ1, τ2, . . . , τk ∈ L each separating u and v. Such a feasible solution is called a land...