For an ordered subset W={w1,w2,…,wk} of vertices and a vertex v in connected graph G, the k-vector r(v|W)=(d(v,w1),d(v,w2),…,d(v,wk)) is called representation with respect to W, where d(v,wi) distance between wi, for 1≤i≤k. The set W resolving G if r(u|W)≠r(v|W), every pair u,v∈V(G). minimum positive integer k which has cardinality metric dimension denoted as dim(G). A dim(G) basis G. bipartite...

An `2 metric is a metric ρ such that √ ρ can be embedded isometrically into R endowed with Euclidean norm, and the minimal possible d is the dimension associated with ρ. A dimension reduction of an `2 metric ρ is an embedding of ρ into another ` 2 2 metric μ so that distances in μ are similar to those in ρ and moreover, the dimension associated with μ is small. Much of the motivation in investi...

A set of vertices W resolves a graph G if every vertex of G is uniquely determined by its vector of distances to the vertices in W . The metric dimension for G, denoted by dim(G), is the minimum cardinality of a resolving set of G. In order to study the metric dimension for the hierarchical product G2 2 uG1 1 of two rooted graphs G2 2 and G u1 1 , we first introduce a new parameter, the rooted ...

It is known that dimension of a set in a metric space can be characterized in information-related terms – in particular, in terms of Kolmogorov complexity of different points from this set. The notion of Kolmogorov complexity K(x) – the shortest length of a program that generates a sequence x – can be naturally generalized to conditional Kolmogorov complexity K(x : y) – the shortest length of a...

An `2 metric is a metric ρ such that √ ρ can be embedded isometrically into R endowed with Euclidean norm, and the minimal possible d is the dimension associated with ρ. A dimension reduction of an `2 metric ρ is an embedding of ρ into another `2 metric μ so that distances in μ are similar to those in ρ and moreover, the dimension associated with μ is small. Much of the motivation in investigat...