An Application of Gd-Metric Spaces and Metric Dimension of Graphs

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

Download Full-text

On the metric dimension and diameter of circulant graphs with three jumps

Discrete Mathematics Algorithms and Applications ◽

10.1142/s1793830918500088 ◽

2018 ◽

Vol 10 (01) ◽

pp. 1850008

Author(s):

Muhammad Imran ◽

A. Q. Baig ◽

Saima Rashid ◽

Andrea Semaničová-Feňovčíková

Keyword(s):

Positive Integer ◽

Connected Graph ◽

Metric Spaces ◽

Metric Dimension ◽

Resolving Set ◽

Circulant Graphs ◽

Infinite Class ◽

Metric Properties ◽

Minimum Number ◽

Metric Basis

Let [Formula: see text] be a connected graph and [Formula: see text] be the distance between the vertices [Formula: see text] and [Formula: see text] in [Formula: see text]. The diameter of [Formula: see text] is defined as [Formula: see text] and is denoted by [Formula: see text]. A subset of vertices [Formula: see text] is called a resolving set for [Formula: see text] if for every two distinct vertices [Formula: see text], there is a vertex [Formula: see text], [Formula: see text], such that [Formula: see text]. A resolving set containing the minimum number of vertices is called a metric basis for [Formula: see text] and the number of vertices in a metric basis is its metric dimension, denoted by [Formula: see text]. Metric dimension is a generalization of affine dimension to arbitrary metric spaces (provided a resolving set exists). Let [Formula: see text] be a family of connected graphs [Formula: see text] depending on [Formula: see text] as follows: the order [Formula: see text] and [Formula: see text]. If there exists a constant [Formula: see text] such that [Formula: see text] for every [Formula: see text] then we shall say that [Formula: see text] has bounded metric dimension, otherwise [Formula: see text] has unbounded metric dimension. If all graphs in [Formula: see text] have the same metric dimension, then [Formula: see text] is called a family of graphs with constant metric dimension. In this paper, we study the metric properties of an infinite class of circulant graphs with three generators denoted by [Formula: see text] for any positive integer [Formula: see text] and when [Formula: see text]. We compute the diameter and determine the exact value of the metric dimension of these circulant graphs.

Download Full-text

BISECTORS IN VECTOR GROUPS OVER INTEGERS

Bulletin of the Australian Mathematical Society ◽

10.1017/s0004972719000807 ◽

2019 ◽

Vol 100 (3) ◽

pp. 353-361 ◽

Cited By ~ 1

Author(s):

SHENG BAU ◽

YIMING LEI

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Metric Dimension ◽

Generating Set ◽

Finite Set ◽

Unit Vectors

We present an example of an isometric subspace of a metric space that has a greater metric dimension. We also show that the metric spaces of vector groups over the integers, defined by the generating set of unit vectors, cannot be resolved by a finite set. Bisectors in the spaces of vector groups, defined by the generating set consisting of unit vectors, are completely determined.

Download Full-text

Metric dimension of a direct sum and direct product of metric spaces

Bulletin of Taras Shevchenko National University of Kyiv. Series: Physics and Mathematics ◽

10.17721/1812-5409.2020/1-2.6 ◽

2020 ◽

pp. 41-46

Author(s):

B Ponomarchuk

Keyword(s):

Direct Product ◽

Metric Space ◽

Metric Spaces ◽

Metric Dimension ◽

Np Hard ◽

Direct Sum

For an arbitrary metric space (X, d) subset A \subset X is called resolving if for any two points x \ne y \in X there exists point a in subset A for which following inequality holds d(a, x) \ne d(a, y). Cardinality of the subset A with the least amount of points is called metric dimension. In general, the problem of finding metric dimension of a metric space is NP–hard [1]. In this paper metric dimension for particular constructs of metric spaces is provided. In particular, it is fully characterized metric dimension for the direct sum of metric spaces and shown some properties of the metric dimension of direct product.

Download Full-text

The Metric Dimension of Metric Spaces

Computational Methods and Function Theory ◽

10.1007/s40315-013-0024-0 ◽

2013 ◽

Vol 13 (2) ◽

pp. 295-305 ◽

Cited By ~ 11

Author(s):

Sheng Bau ◽

Alan F. Beardon

Keyword(s):

Metric Spaces ◽

Metric Dimension

Download Full-text

On the k-metric dimension of metric spaces

Ars Mathematica Contemporanea ◽

10.26493/1855-3974.1281.c7f ◽

2018 ◽

Vol 16 (1) ◽

pp. 25-38 ◽

Cited By ~ 4

Author(s):

Alan F. Beardon ◽

Juan Alberto Rodrı́guez-Velázquez

Keyword(s):

Metric Spaces ◽

Metric Dimension

Download Full-text

Lexicographic metric spaces: Basic properties and the metric dimension

Applicable Analysis and Discrete Mathematics ◽

10.2298/aadm180627004r ◽

2020 ◽

Vol 14 (1) ◽

pp. 20-32

Author(s):

Juan Rodríuez-Velázquez

Keyword(s):

Metric Space ◽

Metric Spaces ◽

Metric Dimension ◽

Basic Properties

In this article, we introduce the concept of lexicographic metric space and, after discussing some basic properties of these metric spaces, such as completeness, boundedness, compactness and separability, we obtain a formula for the metric dimension of any lexicographic metric space.

Download Full-text