### Metric dimension of metric transform and wreath product

#### Abstract

Let $(X,d)$ be a metric space. A non-empty subset $A$ of the set $X$

is called \textit{resolving} set of the metric space $(X,d)$ if

for two arbitrary not equal points $u,v$ from $X$ there exists an element

$a$ from $A$, such that $d(u,a) \neq d(v,a)$. The smallest of cardinalities of

resolving subsets of the set $X$ is called \textit{the metric dimension} $md(X)$ of the metric space $(X,d)$.

In general, finding the metric dimension is an NP-hard problem.

In this paper, metric dimension for metric transform and wreath product of metric spaces

are provided. It is shown that the metric dimension of an arbitrary metric space is equal to the metric dimension of its metric transform.

is called \textit{resolving} set of the metric space $(X,d)$ if

for two arbitrary not equal points $u,v$ from $X$ there exists an element

$a$ from $A$, such that $d(u,a) \neq d(v,a)$. The smallest of cardinalities of

resolving subsets of the set $X$ is called \textit{the metric dimension} $md(X)$ of the metric space $(X,d)$.

In general, finding the metric dimension is an NP-hard problem.

In this paper, metric dimension for metric transform and wreath product of metric spaces

are provided. It is shown that the metric dimension of an arbitrary metric space is equal to the metric dimension of its metric transform.

#### Keywords

metric dimension, metric transform, wreath product

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