The metric dimension of a graph G is the minimum number of vertices in a subset S of the vertex set of G such that all other vertices are uniquely determined by their distances to the vertices in S. In this paper we investigate the metric dimension of the random graph G(n, p) for a wide range of probabilities p = p(n).

SOME RESULTS ON DISCREPANCIES BETWEEN METRIC DIMENSION AND PARTITION DIMENSION OF A GRAPH* Muhammad Imran Centre for Advanced Mathematics and Physics, National University of Sciences and Technology, Sector H-12, Islamabad, Pakistan [email protected] ABSTRACT. In this paper some infinite regular graphs generated by tilings of the plane by infinite hexagonal grid are considered. It is prove...

In the previous lecture notes, we saw that any metric (X, d) with |X| = n can be embedded into R 2 n) under any the `1 metric (actually, the same embedding works for any `p metic), with distortion O(log n). Here, we describe an extremely useful approach for reducing the dimensionality of a Euclidean (`2) metric, while incurring very little distortion. Such dimension reduction is useful for a nu...

A metric basis is a set W of vertices of a graph G(V,E) such that for every pair of vertices u, v of G, there exists a vertex w ∈ W with the condition that the length of a shortest path from u to w is different from the length of a shortest path from v to w. The minimum cardinality of a metric basis for G is called the metric dimension. A pair of vertices u, v is said to be strongly resolved by...

Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let F be a family of connected graphs Gn : F = (Gn)n≥1 depending on n as follows: the order |V (G)| = φ(n) and lim n→∞ φ(n) = ∞ . If there exists a constant C > 0 such that dim(Gn) ≤ C for every n ≥ 1 then we shall say that F has bounded metric dimension. If all graphs in F hav...

In this paper we extend the definitions of mean dimension and metric di-mension for non-autonomous dynamical systems. We show some properties extension furthermore applications to single continuous maps.

Recall that the asymptotic dimension is a coarse geometric analogue of the covering dimension in topology [14]. More precisely, the asymptotic dimension for a metric space is the smallest integer n such that for any r > 0, there exists a uniformly bounded cover C = {Ui}i∈I of the metric space for which the rmultiplicity of C is at most n + 1, i.e. no ball of radius r in the metric space interse...