Research Article
On Constant Metric Dimension of Some Generalized
Convex Polytopes
Xuewu Zuo ,
1
Abid Ali ,
2
Gohar Ali ,
2
Muhammad Kamran Siddiqui ,
3
Muhammad Tariq Rahim ,
4
and Anton Asare-Tuah
5
1
Department of General Education, Anhui Xinhua University, Hefei, China
2
Department of Mathematics, Islamia College, Peshawar, Khyber Pakhtunkhwa, Pakistan
3
Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
4
Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, Khyber Pakhtunkhwa, Pakistan
5
Department of Mathematics, University of Ghana, Legon, Ghana
CorrespondenceshouldbeaddressedtoAntonAsare-Tuah;aasare-tuah@ug.edu.gh
Received 12 June 2021; Accepted 31 July 2021; Published 10 August 2021
AcademicEditor:AntonioDiCrescenzo
Copyright©2021XuewuZuoetal.isisanopenaccessarticledistributedundertheCreativeCommonsAttributionLicense,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Metricdimensionistheextractionoftheaffinedimension(obtainedfromEuclideanspace E
d
)tothearbitrarymetricspace.A
family F �(G
n
) ofconnectedgraphswith n ≥ 3isafamilyofconstantmetricdimensionifdim(G)� k (someconstant)forall
graphsinthefamily.Family F hasboundedmetricdimensionifdim(G
n
) ≤ M,forallgraphsin F.Metricdimensionisusedto
locatethepositionintheGlobalPositioningSystem(GPS),optimization,networktheory,andimageprocessing.Itisalsousedfor
thelocationofhospitalsandotherplacesinbigcitiestotracetheseplaces.Inthispaper,weanalyzedthefeaturesandmetric
dimension of generalized convex polytopes and showed that this family belongs to the family of bounded metric dimension.
1.Introduction
Let G ∈ F be a finite, simple, and undirected connected
graph with vertex set V � V(G)� v
1
,v
2
, ... ,v
n
and edge
set E � E(G).edistancebetweentwoverticesisdenoted
byd(v
s
,v
j
)� d
sj
where d
sj
isthelengthoftheshortestpath
betweentheseverticesin G.Moreover,thedistance d
sj
� d
js
because all graphs are undirected. An ordered subset W �
w
1
,w
2
, ... ,w
k
of V iscalledaresolvingsetorlocatingset
for G ifforanytwodistinctvertices v
s
and v
j
,theircodesare
distinct with respect to Z, where code(v
s
)�
(d(v
s
,z
1
), d(v
s
,z
2
), ... , d(v
s
,z
k
)) ∈ W
k
is a vector [1].
min : |W|: W isaresolvingsetof G � dim(G)� β(G) is
called the metric dimension or locating number of G,and
sucharesolvingset Z iscalledabasissetfor G.Toinvestigate
Z isabasissetfor G,itsufficestoshowthat,foralldifferent
vertices x, y ∈ V∖W, their codes are also different because
forany w
j
∈ W, 1 ≤ j ≤ k,the jthcomponentofthecodeis
zero, while all other components are nonzero. For more
details about β(G) and resolving sets, one can read [1–4].
Lemma 1 (see[3]). For a connected graph G with resolving
set W, if d(x
s
,w)� d(x
j
,w) for all w ∈ V∖ x
s
,x
j
, then
W ∩ x
s
,x
j
≠∅.
ejoinoftwographs G and H representedas G + H is
a graph with V(G + H)� V(G) ∪ V(H) and
E(G + H)� E(G) ∪ E(H) ∪ gh: g ∈ V(G) and h ∈ V(H) .
W
n
� C
n
+ K
1
isawheelgraphoforder n + 1for n ≥ 3. f
n
�
P
n
+ K
1
isafangraphobtainedfromtheamalgamationof
the path on n vertices with a single vertex graph K
n
.
Jahangir or gear graph J
2n
is obtained from the wheel
graph W
2n
by deleting n-cycle edges alternatively; see in
[4]. e following results appear in [5–7] for the graphs
defined above.
Theorem 1. For wheel graph W
n
, fan graph f
n
, and Jahangir
graph J
2n
, we have the following:
(i) β(W
n
) � ⌊(2n + 2)/ 5⌋, for every n ≥ 7
(ii) β(f
n
) � [(2n + 2)/ 5], for every n ≥ 7
Hindawi
Journal of Mathematics
Volume 2021, Article ID 6919858, 7 pages
https://doi.org/10.1155/2021/6919858