Hindawi
Journal of Mathematics
Volume 2021, Article ID 6919858, 7 pages
https://doi.org/10.1155/2021/6919858
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
1Department of General Education, Anhui Xinhua University, Hefei, China
2Department of Mathematics, Islamia College, Peshawar, Khyber Pakhtunkhwa, Pakistan
3Department of Mathematics, COMSATS University Islamabad, Lahore Campus, Pakistan
4Department of Mathematics, Abbottabad University of Science and Technology, Abbottabad, Khyber Pakhtunkhwa, Pakistan
5Department of Mathematics, University of Ghana, Legon, Ghana
Correspondence should be addressed to Anton Asare-Tuah; aasare-tuah@ug.edu.gh
Received 12 June 2021; Accepted 31 July 2021; Published 10 August 2021
Academic Editor: Antonio Di Crescenzo
Copyright © 2021 Xuewu Zuo et al. )is is an open access article distributed under the Creative Commons Attribution License,
which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
Metric dimension is the extraction of the affine dimension (obtained from Euclidean space Ed) to the arbitrary metric space. A
family F � (Gn) of connected graphs with n≥ 3 is a family of constant metric dimension if dim(G) � k (some constant) for all
graphs in the family. FamilyF has bounded metric dimension if dim(Gn)≤M, for all graphs inF. Metric dimension is used to
locate the position in the Global Positioning System (GPS), optimization, network theory, and image processing. It is also used for
the location of hospitals and other places in big cities to trace these places. In this paper, we analyzed the features and metric
dimension of generalized convex polytopes and showed that this family belongs to the family of bounded metric dimension.
1. Introduction Lemma 1 (see [3]). For a connected graph G with resolving
set W, if d(xs, w) � d(xj, w) for all w ∈ V∖􏽮xs, xj􏽯, then
Let G ∈ F be a finite, simple, and undirected connected W∩ 􏽮xs, xj􏽯≠∅.
graph with vertex set V � V(G) � 􏼈v1, v2, . . . , vn􏼉 and edge
set E � E(G). )e distance between two vertices is denoted )e join of two graphs G and H represented as G + H is
by d(vs, vj) � dsj where dsj is the length of the shortest path a graph with V(G + H) � V(G)∪V(H) and
between these vertices in G. Moreover, the distance dsj � djs E(G + H) � E(G)∪E(H)∪ 􏼈gh: g ∈ V(G) and h ∈ V(H)􏼉.
because all graphs are undirected. An ordered subset W � Wn � Cn + K1 is a wheel graph of order n + 1 for n≥ 3. fn �
􏼈w1, w2, . . . , wk􏼉 of V is called a resolving set or locating set Pn + K1 is a fan graph obtained from the amalgamation of
for G if for any two distinct vertices vs and vj, their codes are the path on n vertices with a single vertex graph Kn.
distinct with respect to Z, where code(vs) � Jahangir or gear graph J2n is obtained from the wheel
(d(vs, z1), d(vs, z2), . . . , d(vs, zk)) ∈Wk is a vector [1]. graph W2n by deleting n-cycle edges alternatively; see in
min : 􏼈|W|: W is a resolving set of G􏼉 � dim(G) � β(G) is [4]. )e following results appear in [5–7] for the graphs
called the metric dimension or locating number of G, and defined above.
such a resolving set Z is called a basis set for G. To investigate
Z is a basis set for G, it suffices to show that, for all different Theorem 1. For wheel graph Wn, fan graph f , and Jahangirvertices x, y ∈ V∖W, their codes are also different because ngraph J2n, we have the following:for any wj ∈W, 1≤ j≤ k, the jth component of the code is
zero, while all other components are nonzero. For more (i) β(Wn) � ⌊(2n + 2)/5⌋, for every n≥ 7
details about β(G) and resolving sets, one can read [1–4]. (ii) β(fn) � [(2n + 2)/5], for every n≥ 7
2 Journal of Mathematics
(iii) β(J2n) � ⌊2n/3⌋, for every n≥ 4 generalized convex polytopes with pendent edges for their
metric dimensions.
All the above three families of graphs are planar, and
their metric dimension depends on the number of vertices 2. Main Results
in the graph, which shows that the metric dimension of
these graphs is unbounded [8, 9]. Khuller et al. [10] clarified )is section is devoted to the main results which we proved
the properties of those graphs whose metric dimension is for the newly introduced generalized convex polytopes. )e
two. convex polytopes Sn, Tn, and Un were examined by
Muhammad et al. for their metric dimensions in [2] and
Theorem 2 (see [10]). Let β(G) � 2 � |W|, where proved that these families belong to the family of constant
W � 􏼈x, y􏼉 ⊂ V(G); then, the following holds: metric dimension.
Generalized convex polytope Sn,m is the generalization of
(i) 7ere is a unique shortest path P between x and y Sn, with one n-sided and infinite face each, 3-sided faces being
(ii) deg(x)≤ 3 or deg(y)≤ 3 2n, and 4-sided faces being n(m − 2), so the total number of
(iii) For every other vertex z except x and y on P, faces is nm + 2. )e convex polytope
p
Sn,m is obtained from the
deg(z)≤ 5 generalized convex polytope graph by attached p-pendent
vertices at the outer cycle of Sn,m, shown in Figure 1. )e
generalized convex polytope pSn.m with p-pendents is a graph
Definition 1 (see [11]). A set K ⊂ Rd is said to be convex if consisting of m cycles, with vertex and edge sets
the line segment xy: λx + (1 − λ)y, 0≤ λ≤ 1, lies inside K p j
for all distinct pairs of point x, y ∈ K. V􏼐Sn,m􏼑 � 􏽮Xs : 1≤ s≤ n, 1≤ j≤m􏽯,
p j j
E􏼐S 􏼑 � 􏽮X Xs+1: 1≤ s≤ n, 1≤ j≤m􏽯
Definition 2 (see [11]). )e smallest convex set containing n,m sK (1)
(the intersection of the family of all convex sets that contain ∪ j j+1􏽮XsXs : 1≤ s≤ n, 1≤ j≤m􏽯
K) is called the convex hull of K, denoted by 1 2
Conv(K) � ∩ K⊂S Ss, where Ss is a convex set.
∪ 􏽮Xs+1Xs : 1≤ s≤ n􏽯.
s
In the set of edges, indices are taken as modulo n and m.
Definition 3 (see [11]). A convex polytope is a bounded In [2], it was shown that β(Sn) � 3, for n≥ 6. In the resultconvex linear combination of convex sets. below, we proved that the metric dimension for the gen-
eralized convex polytope of Sn is still 3, which implies that S ,)ere are some families of graphs with constant metric npSn , and
p
Sn,m belong to the same family of constant metric
dimension (see [2]); these families are generated by convex dimension.
polytopes. )e problem of finding β(G) is NP-complete (see
[2]). Theorem 6. Let pG � Sn,m be the generalized convex polytope
graph defined above; then, β(G) � 3 for n≥ 6 and m≥ 5.
Theorem 3 (see [12]). Let pSn be a convex polytope with
p-pendent vertices; then, dim p(Sn ) � 3 for all n≥ 6. Proof. Validating the mentioned theorem with the help of
double inequalities, two cases are present:
Theorem 4 (see [12]). 7e metric dimension of convex
polytope p with -pendent edges is 3 for every ≥ 6. Case (i): for n is even.Tn p n
Let n � 2α′ where α′ ≥ 3 is an integer. As |N2(x)|≥ 6 for
Theorem 5 (see [12]). β p(U ) � 3 for n≥ 6, where pU is a all x ∈ Sn,m, it is guaranteed by [15] that β(G)≥ 3.n n
convex polytope graph with p-pendents. Consider Z � 􏼈X11, X
1
2, X
1
l+1􏼉 to be an ordered subset of
p
V(Sn,m); to show that Z is a basis set for G, codes of the
For more details about the metric dimension of certain elements of pV(Sn,m)∖Z with respect to Z are given in
families of graphs, see [13, 14]. Here, we will investigate the following scheme:
1 ⎧⎨ ( s − 1, s − 2, α′ − s + 1􏼁, α′ ≥ s≥ 3,
r􏼐Xs |Z􏼑 � ⎩
( 2α′ − s + 1, 2α′ − s + 2, s − α′ − 1􏼁, 2α′ ≥ s≥ α′ + 2,
⎧⎪ ( 1, 1, α′􏼁, s � 1,
⎪ (2)
⎪
2 ⎨ ( s, s − α′, α′ − s + 1􏼁, α′ ≥ s≥ 2,
r􏼐Xs |Z􏼑 � ⎪
⎪ ( α′, α′, 1􏼁, s � α′ + 1,
⎪⎩
( 2α′ − s + 1, 2α′ − s + 2, s − l􏼁, 2α′ ≥ s≥ α′ + 2.
Journal of Mathematics 3
Xm1
Xm-1
Xm 12
Xm-1 X
m
n
2
Xm-1n
4
Xm X4
X1
3 Xm-1 2 3 X
m
3 X3 X1 X4 n-13 n
X2 X
3
n Xm-1X4 1 n-13 X3 23 X2
X2 X21
3 X X1 n2 1 4
X1 X1 3 Xn 2 Xn-1 n-13 X
X1 1 n-14 Xn-1
Figure 1: )e generalized convex polytope graph pSn,m.
Codes for the vertices Xms for 1≤ s≤ n and m≥ 3 are Let n � 2α′ + 1, where α′ ≥ 3, and by [15], β(G)≥ 3; for
given in the following: reaching the conclusion, it remains to show that
m β(G)≤ 3.
r( Xs |Z􏼁 � (m − 2, m − 2, m − 2
2
) + r􏼐Xs |Z􏼑. (3) Let Z � 􏼈X11, X12, X1l+1􏼉 be an ordered subset of
p
Sn,m; the
It proves that β ≤ 3 implies that the metric di- formulation for the representation of nodes for(G) p
mension of p is 3. V(Sn,m)∖Z with respect to Z is given in the following:G � Sn,m
Case (ii): for n is an odd integer.
⎧⎪ ( s − 1, s − 2, α′ − s + 1􏼁, α′ ≥ s≥ 3,
⎪
1 ⎨
r􏼐Xs |Z􏼑 � ⎪ ( α′, α′, 1􏼁, s � α′ + 2,⎪
⎩
( 2α′ − s + 2, 2α′ − s + 3, s − α′ − 1􏼁, 2α′ + 1≥ s≥ α′ + 3,
⎧⎪⎪ (
1, 1, α′􏼁, s � 1, (4)
⎪
2 ⎨ ( s, s − α′, α′ − s, l − s + 1􏼁, α′ ≥ s≥ 2,
r􏼐Xs |Z􏼑 � ⎪⎪
⎪ ( α′ + 1, α′, 1􏼁, s � α′ + 1,
⎩
( 2α′ − s + 2, 2α′ − s + 3, s − α′􏼁, n≥ s≥ α′ + 2.
p j
V􏼐T 􏼑 � 􏽮X : 1≤ s≤ n, 1≤ j≤m􏽯,
)e representation of the vertices Xms , 1≤ s≤ n and
n,m s
≥ 3, is as follows: p j
j
m E􏼐Tn,m􏼑 � 􏽮XsXs+1: 1≤ s≤ n, 1≤ j≤m􏽯
∪ j j+1m 2 􏽮X X : 1≤ s≤ n, 1≤ j≤m − 3􏽯
r( Xs |Z􏼁 � (m − 2, m − 2, m − 2) + r􏼐Xs |Z􏼑. (5) s s
∪ m−2 m−1 m−1 m􏽮Xs+1 Xs : 1≤ s≤ n􏽯∪ 􏽮Xs Xs : 1≤ s≤ n􏽯.
It shows that, for any two distinct vertices x, y ∈ pSn,m for (6)
odd n≥ 7, r(x|Z)≠ r(y|Z) implying that β(G)≤ 3; this
completes the proof. □ In the set of edges, indices are taken as modulo n and m.
In Figure 2, the graph pG � Tn,m is shown.
)e result given below shows that pTn,m belongs to the
3. Generalized Convex Polytope Graph pTn,m family of constant metric dimension.
In [16], Imran et al. proved the metric dimension of convex p
polytope T . )e general form of T is denoted by T Theorem 7. Let Tn,m be a GCP graph with p-pendents for alln n n,m p
known as the generalized convex polytope (for short, GCP); n≥ 6; then, β(Tn,m) � 3.
this graph consists of one each n-sided and infinite face,
respectively, and the number of 3-sided faces is 4 pn and 4- Proof. As |N2(v)|≥ 6 for all nonpendent vertices of Tn,m,
sided faces is n(m − 3). )e GCP graph p pTn,m is a graph with β(Tn,m)≥ 3 for all n≥ 6 by [15]. To complete the proof, it
p-pendent edges. Vertex and edge sets for pG � Tn,m are given suffices to show that any ordered subset of the vertices of this
in the following: graph is a resolving set.
4 Journal of Mathematics
Case (i): for is an even integer. vertices of pn V(Tn,m)∖Z with respect to Z is formulated
Let n � 2α′ with α′ ≥ 3; consider an ordered subset Z � as follows:
p
􏼈X11, X
1
2, X
1
l+1􏼉 of vertices of Tn,m. )e representation of
1 ⎧⎨ ( s − 1, s − 2, α′ − s + 1􏼁, α′ ≥ s≥ 3,
r􏼐Xs |Z􏼑 � ⎩
( 2α′ − s + α′, 2α′ − s + 2, s − α′ − 1􏼁, 2α′ ≥ s≥ α′ + 2,
⎧⎪ ( 1, 1, α′􏼁, s � 1,
⎪⎪ (7)
2 ⎨ ( s, s − α′, α′ − s + 1􏼁, α′ ≥ s≥ 2,
r􏼐Xs |Z􏼑 � ⎪
⎪ ( α′, α′⎪ , 1􏼁, s � α′ + 1,
⎩
( 2α′ − s + 1, 2α′ − s + 2, s − α′􏼁, 2≥ s≥ α′ + 2.
For 3≤ j≤m − 2,
j 2
r􏼐Xs |Z􏼑 � (j − 2, j − 2, j − 2) + s􏼐Xs |Z􏼑. (8)
⎧⎪⎪ ( 2 + 1, 2 + 1, α′ + 1􏼁, s � 1,
⎪
⎪ ( s + 2, s + 1, α′ − s + 2􏼁, α′ − 1≥ s≥ 2,
⎪
⎪ ( α′ + 2, α′⎪ + 1, 3􏼁, s � α′,⎨
m−1
r􏼐X |Z􏼑 � ⎪ ( α′s + 1, α′ + 2, 3􏼁, s � α′ + 1, (9)
⎪
⎪ ( 2α′ − s + 2, 2α′ − s + 3, s − α′ + 2􏼁,
⎪⎪
⎪ α′ + 2≤ s≤ n − 1,
⎩⎪
( 3, 3, α′ + 2􏼁, s � n.
Codes of the pendent vertices are given as follows: Case (ii): for n is an odd integer.
m m−1
r( X |Z􏼁 � (1, 1, 1) + r􏼐X |Z􏼑. (10) Let n � 2α′ + 1 for α′ ≥ 3; suppose an ordered subsets s
Z � 􏼈x1, X1, X1
p
1 2 l+1􏼉 of vertices V(Tn,m); to show that Z is
p
From the above formulation, it is obvious that no two a basis set for Tn,m, the formulation codes are given as
distinct vertices of the GCP with pendents p have the follows:
same code with respect to Z, which implies that
β p(Tn,m) � 3.
⎪⎪⎧ ( s − 1, s − 2, α′ − s + 1􏼁, α′ ≥ s≥ 3,
1 ⎨
r􏼐X |Z􏼑 � ⎪ ( α′, α′, 1􏼁, s � α′s + 2,
⎩⎪
( 2α′ − s + 2, 2α′ − s + 3, s − α′ − 1􏼁, n≥ s≥ α′ + 3,
⎧⎪ ( 1, 1, α′􏼁, s � 1, (11)
⎪⎪
2 ⎨ ( s, s − 1, α′ − s + 1􏼁, α′ ≥ s≥ 2,
r􏼐Xs |Z􏼑 � ⎪
⎪ ( α′ + 1, α′, 1􏼁, s � α′⎪ + 1,
⎩
( 2α′ − s + 2, 2α′ − 3, s − α′ − 1􏼁, n≥ s≥ α′ + 2.
Journal of Mathematics 5
Codes given to the vertices of other interior cycles are
j 2
r􏼐Xs |Z􏼑 � (j − 2, j − 2, j − 2) + r􏼐Xs |Z􏼑, for 1≤ j≤m − 2. (12)
Representation given to the second last cycle Xm−1s is
⎪⎧ ( 2 + 1, 2 + 1, α′ + 1􏼁, s � 1,
⎪
⎪
⎪ ( s + 2, s + 1, α′ − s + 2􏼁, α′ − 1≥ s≥ 2,
⎪
⎪
⎪ ( α′ + 2, α′ + 1, 3􏼁, s � α′,
m−1 ⎨
r􏼐Xs |Z􏼑 � ⎪ ( α′ + 2, α′ + 2, 3􏼁, s � α′ + 1, (13)⎪
⎪
⎪ ( 2α′ − s + 3, 2α′ − s + 4, s − α′ + 2􏼁,
⎪
⎪
⎪ ( n − 1≥ s≥ α′ + 2􏼁,
⎪⎩
( 3, 3, α′ + 2􏼁, s � n.
)e same representation is given to the pendent vertices: We will show that GCP graph pUn,m with n≥ 6 along with
m m−1 p-pendent vertices belongs to the family of constant metric
r( Xs |Z􏼁 � (1, 1, 1) + r􏼐Xs |Z􏼑. (14) dimension and its locating number is 3.
It shows that is a resolving set for pZ Tn,m for n-odd and
p-pendents, β p(Tn,m) � 3, and this completes the proof. □ Theorem 8. Let pUn,m be a GCP graph for n≥ 6; then,
dim p(Un,m) � 3.
4. Generalized Convex Polytope Graph pUn,m
In [2], the graph Un is given, and a generalized graph
p
Un,m of Proof. According to [15], dim(G)≥ 3 if and only if
Un is shown in Figure 3. )e vertex and edge sets for this |N2(x)|≥ 6 or |N3(x)|≥ 8 for all x ∈ V(G) as |N2(x)|≥ 6 forp
graph are given as follows: every nonpendent vertex x of Un,m implying that
dim p(U )≥ 3. To reach the conclusion, it remains to show
p j n,m
V􏼐Un,m􏼑 � 􏽮Xs : 1≤ s≤ n, s≤ j≤m􏽯, that there exists a resolving set for pUn,m with exactly three
p j j 1≤ ≤ 1≤ ≤ 1 elements. For this, consider the following two cases:E􏼐Un,m􏼑 � 􏽮XsXs+1: s n, j m − 􏽯 Case (i): for an integer n is even. Let n � 2α′, where
∪ j j+1􏽮XsXs : 1≤ s≤ n, 1≤ ≤
1 1 1
j m − 4􏽯 α′ ≥ 3; take Z � 􏼈X1, X2, Xl+1􏼉 to be an ordered subset of
p
3 2 V(Un,m); to show that Z resolves vertices of the GCP, the∪ m− m−􏽮Xs Xs : 1≤ s≤ n􏽯 (15) representation of vertices of the GCP is shown as follows:
∪ m−2 m−3􏽮Xs Xs+1 : 1≤ s≤ n􏽯
∪ m−2 m−1􏽮Xs Xs : 1≤ s≤ n􏽯
∪ m−1 m􏽮Xs Xs : 1≤ s≤ n􏽯.
1 ⎧⎨ ( s − 1, s − 2, s − α′ + 1􏼁, α′ ≥ s≥ 3,
r􏼐Xs |Z􏼑 � ⎩ (16)
( 2α′ − s + 1, 2α′ − s + 2, s − α′ − 1􏼁, 2α′ ≥ s≥ α′ + 2.
6 Journal of Mathematics
Xm1 Xmn
m-1 X
m-2
X 11 m-1
Xm m-2
Xn
2 X2 Xm-2n Xmn-1
Xm-12 4 Xm-1n-1
X4 X12
3 X3m-2 X 1 X3 X4 m-2X3 2 n
n Xn-1
X4 2 X2 X
3
3 3 X2 1 X2 n2 1 n Xm
Xm-1
X3 X3 X2 X1 4 n-11 1
Xm
3 X3 X1
3 X
2X n-1 X
3 1 1n
X n-1
X n-14 Xn-1
X 2
Xm-2 X4 X4 X4 Xn-24 X
3
n-2X4n-2 Xm-2n-2
Figure 2: GCP graph pTn,m.
Xm1 m
Xm-1
Xn
1
Xm-1n
Xm-3
Xm m-2
1
2 X m-2Xm-1 Xm-3 1 X2 2 n Xm-3n Xmn-1
Xm-2 Xm-12 Xm-2 n-13 n-13 X
m-3 X2 1X3 X2 X2 3 m-3m-1 3 2 1
Xn Xn-1
X3 X3 2 mm-2 X1 X1 Xn X2 1 1 m-2 n-2
Xm
X3 2 X1 X 3 Xn
3 X3 3 1 X
2 Xn-1 n-2 X
X n-1
Xm-3 X X
1 n-1
4 Xm-34 X 4 n-24 X2n-2 Xn-2
Figure 3: )e generalized convex polytope graph pUn,m.
For 2≤ j≤m − 3, Representation given to the Xm−2i cycle is
j 1
r􏼐Xs |Z􏼑 � (j − 1, j − 1, j − 1) + r􏼐Xs |Z􏼑. (17)□
⎪⎧ ( 3, 3, α′ + 2⎪ 􏼁, s � 1
⎪
2 ⎨ ( s + 2, s + 1, α′ − s + 3m− 􏼁, α′ ≥ s≥ 2,
r􏼐Xs |Z􏼑 � ⎪ (18)⎪ ( α′ + 2, α′⎪ + 2, 3􏼁, s � α′ + 1,
⎩
( 2α′ − s + 3, 2α′ − s + 4, s − α′ + 2􏼁, 2α′ ≥ s≥ α′ + 2.
Representation of the vertices of the outer cycle is It shows that Z is a resolving set for GCP pUn,m implying
m−1 m−2 that dim
p
(Un,m) � 3.
r􏼐Xs |Z􏼑 � (1, 1, 1) + r􏼐Xs |Z􏼑, 1≤ s≤ n. (19) Case (ii): when n is an odd integer. Let n � 2α′ + 1 for
1
Representation of pendent vertices is α′ ≥ 3; let Z � 􏼈X1, X
1
2, X
1
l+1􏼉 be an ordered subset of the
vertices of the GCP. To show that Z is a locating set for pUn,m,
m m−2
r( Xs |Z􏼁 � (2, 2, 2) + r􏼐Xs |Z􏼑, 1≤ s≤ n. (20) consider the codes’ formulation of the vertices of the GCP
with respect to Z as
⎪⎧ ( s − 1, s − 2, α′ − s + 1􏼁, α′ ≥ s≥ 3,
⎨⎪1
r􏼐X1|Z􏼑 � ⎪ ( α′, α′, 1􏼁, s � α′ + 2, (21)
⎪⎩
( 2α′ − s + 2, 2α′ − s + 3, s − α′ − 1􏼁, 2α′ + 1≥ s≥ α′ + 3.
Journal of Mathematics 7
For 2≤ j≤m − 2, Representation given to the vertices of the interior cycle
j 1 is
r􏼐Xs |Z􏼑 � (j − 1, j − 1, j − 1) + r􏼐Xs 􏼑. (22)
⎧⎪ ( 3, 3, α′ + 2􏼁, s � 1,
⎪
⎪
⎪ ( s + 2, s + 1, α′ − s + 3􏼁, α′ ≥ s≥ 2,
⎪
⎪
⎪ ( α′ + 3, α′ + 2, 3􏼁, s � α′ + 1,
m−2 ⎨
r􏼐Xs |Z􏼑 � ⎪ ( α′ + 2, α′ + 1, 4􏼁, s � α′ + 2, (23)
⎪⎪
⎪ ( α′ + 1, α′, 5􏼁, s � α′ + 3,
⎪⎪
⎪ ( 2α′ − s + 4, 2α′ − s + 5, s − α′ + 2􏼁,
⎪
⎩ 2α′ + 1≥ s≥ α′ + 3.
Representation given to the nodes of the outer cycle is [4] I. Tomescu, J. Imran, and S. Imran, “On the partition and
1 2 connected partition dimension ofWheels,”Ars Combinatoria,m− m−
r􏼐Xs |Z􏼑 � (1, 1, 1) + r􏼐Xs |Z􏼑, 1≤ s≤ n. (24) vol. 84, pp. 311–318, 2007.
[5] B. Peter, C. Gary, C. Poisson, and P. Zhang, “On k-dimen-
Also, representation given to the nodes hanging is sional graphs and their bases,” Periodica Mathematica
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