Research Article
Computation of Metric Dimension of Certain Subdivided
Convex Polytopes
S.Imran,
1
Z.Ali,
2
N.Nigar,
2
SyedAjazK.Kirmani,
3
M.K.Siddiqui,
4
andS.A.Fufa
5
1
Govt. KRS. College, Walton Road, Lahore, Pakistan
2
Department of Mathematics, Minhaj University Lahore, Lahore, Pakistan
3
Department of Electrical Engineering, College of Engineering, Qassim University, Unaizah, Saudi Arabia
4
Department of Mathematics, Comsats University Islamabad, Lahore Campus, Lahore, Pakistan
5
Department of Mathematics, Addis Ababa University, Addis Ababa, Ethiopia
Correspondence should be addressed to S. A. Fufa; samuel.asefa@aau.edu.et
Received 10 January 2022; Accepted 8 February 2022; Published 8 March 2022
Academic Editor: Gul Rahmat
Copyright©2022S.Imranetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,which
permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
e distance d(z
1
,z
2
) from vertex z
1
∈ V(G) to z
2
∈ V(G) is minimum length of (z
1
,z
2
)-path in a given connected graph G
having E(G) and V(G) edges and vertices’/nodes’ sets, respectively. Suppose Z � z
1
,z
2
,z
3
, ... ,z
m
⊆V(G) is an order set and
c ∈ V(G), and the code of c with reference to Z is the m-tuple {d(c, z
1
), d(c, z
2
), d(c, z
13
), ..., d(c, z
k
)}. en, Z is named as the
locating set or resolving set if each node of G has unique code. A locating set of least cardinality is described as a basis set for the
graph G, and its cardinal number is referred to as metric dimension symbolized by dim(G). Metric dimension of certain
subdivided convex polytopes ST
n
has been computed, and it is concluded that just four vertices are sufficient for unique coding of
all nodes belonging to this family of convex polytopes.
1.Introduction
In the discipline of computer science and mathematics,
graph theory [1] is the survey of graphs that considers the
link between edges and vertices. is is the most celebrated
discipline these days that has applications [2] in computer
science, information technology, biosciences, mathematics,
social sciences, physics, chemistry, and linguistics. To il-
lustrate pairwise relationship of objects, graph theory
analysis is very important [3, 4].
Formally, a graph is the collection of vertices and edges.
Among several types of different graphs, we will analyze a
particular class of graph known as convex polytopes [5].
Convex polytopes are the principal geometric structures
which are under investigation since antiquity. e charm of
this concept is nowadays complemented by their signifi-
cance for various mathematical fields, extending from al-
gebraic geometry, linear programming, integration, and
combinatorial optimization. Convex polytope is the simplest
kind of polytopes [6] which satisfies the property of convex
set in k-dimensional Euclidean space R
k
. Convex polytopes
play a vital part in enormous areas of mathematics as well as
in applied disciplines, but its role in linear programming is
most influential [7, 8].
Moreover, subdividing is a process in which we add an
extra vertex on each edge of the graph in such a way such
that each will be splitted into two edges, and the resulting
graph is called subdivided graph of the original graph G.
Since a couple of years, the variables associated with
distances in graphs have enchanted the focus of various
researchers, but in the recent years, the phenomenon that
has centered certain surveys is termed as metric dimen-
sion [9]. e distance d(z
1
,z
2
) from vertex z
1
∈ V(G) to
z
2
∈ V(G) is minimum length of (z
1
,z
2
)-path in a given
connected graph G having E(G) and V(G) edges and
vertices’/nodes’ sets, respectively. Suppose Z � z
1
,
z
2
,z
3
, ... ,z
m
}⊆V(G) isanordersetand c ∈ V(G);thecode
of c with reference to Z is the m-tuple {d(c, z
1
), d(c, z
2
),
d(c, z
13
), ..., d(c, z
k
)}. en, Z is named as the locating set
or resolving set if each node of G has a unique code. A
Hindawi
Journal of Mathematics
Volume 2022, Article ID 3567485, 9 pages
https://doi.org/10.1155/2022/3567485