Sharp bounds for partition dimension of generalized Möbius ladders

Zafar Hussain; Junaid Alam Khan; Mobeen Munir; Muhammad Shoaib Saleem; Zaffar Iqbal

doi:10.1515/math-2018-0109

Enjoy 40% off

academic books on De Gruyter Brill *

Article Open Access

Sharp bounds for partition dimension of generalized Möbius ladders

Zafar Hussain , Junaid Alam Khan , Mobeen Munir , Muhammad Shoaib Saleem and Zaffar Iqbal

Published/Copyright: November 8, 2018

Published by

Become an author with De Gruyter Brill

Submit Manuscript Author Information Explore this Subject

$Open Mathematics$

From the journal Open Mathematics Volume 16 Issue 1

Abstract

The concept of minimal resolving partition and resolving set plays a pivotal role in diverse areas such as robot navigation, networking, optimization, mastermind games and coin weighing. It is hard to compute exact values of partition dimension for a graphic metric space, (G, d_G) and networks. In this article, we give the sharp upper bounds and lower bounds for the partition dimension of generalized Möbius ladders, M_{m, n}, for all n≥3 and m≥2.

Keywords: Metric dimension; Partition dimension; Generalized Möbius Ladder

MSC 2010: 05C12; 05C15; 05C78

1 Introduction

Computer networks can be modeled on the grounds of graphs, where hosts, servers or hubs can be considered as vertices and edges – as connecting medium between them. Vertex is actually a possible location to find a fault or some damaged devices in a computer network. This idea somehow urged Slater and independently Harary and Meletr in [1] to uniquely recognize each vertex of a graph in a network so that a fault could be controlled in an efficient way. Thus, the basis for notion of locating sets and locating number of graphs came into existence. Since then, the resolving sets have been investigated a lot [1]. The resolving set contributes in various areas such as connected joins in graphs [2], network discovery [3 – 5], strategies for the mastermind games [3, 4], applications of pattern recognition, combinatorial optimization, image processing [6], pharmaceutical chemistry and game theory.

Consider a simple, connected graph G, and metric d_G:V(G) × V(G) → ℕ∪0, where ℕ is the set of positive integers and d_G(x, y) is the minimum number of edges in any path between x and y. Let W = {w₁, w₂,...,w_k} be an ordered set of vertices of G and let v be a vertex of G. The representation r(v|W) of v with respect to W is the k−tuple (d(v, w₁),d(v, w₂),...,d(v, w_k)). If distinct vertices of G have distinct representation with respect to W, then W is called a resolving set of G, see [1]. Such resolving set with minimum cardinality is a basis of G and metric dimension of G, denoted by dim(G) is its cardinality, [7, 8].

Buczkowski et al. established metric dimension of wheel W_n to be ⌊2n+25⌋ for n ≥ 7 [9], Caceres et. al. [10] found that the metric dimension of fan is ⌊2n+25⌋ for n ≥ 7 and Tomescu et. al. [11] determined the dimension of Jahangir graphs J_2n to be ⌊2n3⌋ for all n ≥ 4.

A particular metric-feature of the family of graphs is independence of metric dimension on the particular element of the family. A connected graph has constant metric dimension if dim(G) = k where k ∈ Z⁺. In [8] Chartrand et. al. proved that a graph has constant metric dimension 1 iff it is a path. In [12] the authors discussed some families of constant meric dimensions. The authors computed metric dimension of wheels in [13] and uni-cyclic graphs in [14]. The authors in [15] computed metric dimension of alpha boron nanotubes. Javaid et. al. computed metric dimension of P(n, 3) and established new results on metric dimension of rotationally-symmetric graph. Murtaza et. al. computed partial results of metric dimension of Möbius ladder in [16] whereas Munir et. al. computed exact and complete results for metric dimension of Möbius Ladders in [17].

A variant of metric dimension of a connected graph is a partition dimension of graph introduced in [19, 20, 21, 22, 23] given as : Let G be a connected graph, a subset S ⊂ V(G) and a vertex v, distance d(v, S) = min{d(v, x):x ∈ S}. If Π = {S₁,...S_t} is an ordered t-partition of V(G), then r(v|Π) = {d(v, S₁),...,d(v, S_t)}is the t-tuple representation of v with respect to Π. If this t-tuple representation of v, r(v|Π)for all v ∈ V(G) being all distinct, then this Π is called a resolving partition and the minimum cardinality of such resolving partition is a partition dimension, represented as pd(G).

A natural question may be asked: are partition dimension and metric dimension related in some way? In [20, 21], Cartrand et. al. proved that pd(G) ≤ β(G) + 1 for a non-trivial connected graph G. But in [22, 23], Tomescu et. al. proved that it can be much smaller than the metric dimension. In fact, the authors completed the list of all 23 examples of connected graphs of order n having partition dimensions 2, n − 1 or n. They also gave an example of graphs with finite partition dimension but those which have infinite metric dimension. Recently, Hernando et. al. has proved that there are only 15 families of such type. Tomescu et. al. computed the bounds for the partition dimension of wheel graph in [23]. In [24], the authors computed some bounds for metric and partition dimension of a connected graph. In [25], the authors obtained some sharp bounds for the partition dimension of unicyclic graphs.

Chartrand et al. proved in [22] that if G is a connected graph of order n ≥ 2 then pd(G) = 2 if and only if G is a path, pd(G) = n if and only if G=K_nand for n ≥ 5 pd(G) = n − 1 if and only if G is one of the graphs K_{1,n − 1}, K_n − e, K₁ + (K1∪K_{n + 2}). In [22] Tomescu and Imran studied infinite regular graphs which are generated by tailings of the plane by regular triangles and hexagons. They proved that these graphs have no finite metric bases but their partition dimension is finite and they evaluated this dimension in some cases. In [23], they computed a partition dimension and a connected partition dimension of wheel graphs and showed that n≥4,⌈(2n)13⌉≤pd(G)≤2⌈n12⌉+1. The following lemma gives a general upper bounds for the partition dimension of a graph of size n.

Lemma 1.1.

If ∣G∣ ≥ 3, then pd(G) ≤ n − diam(G) + 1

In this article we want to compute sharp bounds for partition dimension of Generalized Möbius ladders.

2 Generalized Möbius ladders

The classical Möbius ladder M_n is a cubic circulant graph with an even number of vertices, formed from an n-cycle by adding edges connecting opposite pair of vertices in the cycle, except with two pairs which are connected with a twist, as you can see in the figure:

$Fig. 1 Möbius ladder M16$

Fig. 1

Möbius ladder M₁₆

This graph has been an active area of research. For instance, [16, 17] give complete results for its metric dimension. In [26] the authors computed a distance labeling of this graph and also introduced its generalization referred to as Möbius ladder. In [27], the authors not only redefined this generalization in a novel way but also computed metric dimension of M_{m, n}. They also obtained the results of [16, 17] as easy consequences of the results in~[27]. Consider the Cartesian product P_m × P_n of paths P_m and P_n with vertices u₁, u₂,…,u_m and v₁, v₂,…,u_n, respectively. Take a 180_o twist and identify the vertices (u₁, v₁),(u₁, v₂),…,(u₁, v_n) with the vertices (u_m, v_n), (u_m, v_{n − 1}), …,(u_m, v₁), respectively, and identify the edge ((u₁, i), (u₁, i + 1)) with the edge ((u_m, v_{n + 1 − i}), (u_m, v_{n − i})), where 1 ≤ i ≤ n − 1. What we receive is the generalized Möbius ladder M_{m, n}. You may observe that we receive the usual Möbius ladder for n = 2 and for any odd integer m ≥ 4. You can see M_7,3 in the following figure.

$Fig. 2 P7$

Fig. 2

P₇

$Fig. 3 P3$

Fig. 3

P₃

$Fig. 4 P7 × P3$

Fig. 4

P₇ × P₃

For brevity we shall use the symbol v_ij (or simply ij) to represent the vertex (u_i, v_j) of M_{m, n}, as you can see in the figure:

$Fig. 5 P7 × P3 with complete simple labels$

Fig. 5

P₇ × P₃ with complete simple labels

The generalized Möbius ladder obtained from P₇ × P₃ is:

$Fig. 6 M7,3$

Fig. 6

M_7,3

So the generalized Möbius ladder M_{m, n} is a non-regular simple connected graph on n(m − 1) vertices. This article deals with the computation of sharp upper bounds and lower bounds for partition and metric dimensions of M_{m, n}.

3 Main results and discussions

In this part we give our main results. We begin with the sharp upper bounds for the partition dimension of M_{m, n}. Then we move towards the lower bounds.

Theorem 3.1.

For m ≥ 3 and n ≥ 2

3≤pd(Mm,n)≤5,whenn≡1(mod2)andm≡1(mod2),m−n≥44,whenn≡0(mod2)andm≡1(mod2)4whenn≡1(mod2)andm≡0(mod2)5whenn≡0(mod2)andm≡0(mod2),m−n≥4

At first we compute the upper bounds. We construct a general resolving partition on a case by case basis.

3.1 Upper bound

Proof

We divide the proof in two cases on the basis of parities of m and n.

Case I. When m and n are of opposite parity

Let Π = {S₁, S₂, S₃, S₄} Where S₁ = {V_1,1}, S₂ = {V_1,n}, S₃ = {V_1,2, V_1,3,...,

V_{1,n − 1}, V_2,1, V_2,2,......,V_2,n,.....,V_{m − 2,1}, V_{m − 2,2},...,

V_{m − 2,n}, V_{m − 1,2}, V_{m − 1,3},...,V_{m − 1,n}} S₄ = {V_{m − 1,1}}. We prove that Πis a resolving partition for M_{m, n}. To find distance vectors we use two parameters q, i and depending on their different values we divide the entries of distance vectors into four steps.

Step I: Distances of S₁ with all vertices of M_{m, n}.

In this case for each value of q ∈ {1,2,...,n} the parameter i varies from 1 to m − 1. The entries of different vectors are

d(S1,Vi,q)=i+q−2,1≤i≤12(m+n−2q+1)m+n−q−i,12(m+n−2q+3)≤i≤m−1

Step II: Distances of S₂ with all vertices of M_{m, n}.

For each value of q ∈ {1,2, ..., n} the parameter i varies from 1 to m - 1 and we get d(S₂, V_{i, q}) = d(S₁, V_{i, n + 1 − q}).

Step III : Distances of S₃ with all vertices of M_{m, n}.

Here for each value of q ∈ {1,2,...,n} the parameter i varies from 1 to m - 1 and we have

d(S3,Vi,q)=1,ifi=1,q=1,q=n1,ifi=m−1,q=n0,otherwise

Step IV: Distances of S₄ with all vertices of M_{m, n}.

Here we have two parts

a) For q = 1 , we have

d(S4,Vi,q)=i+n−1,1≤i≤12(m−n−1)m−1−i,12(m−n+1)≤i≤m−1

b) For each value of q ∈ {2,..., n} the parameter i varies from 1 to m - 1 and we have d(S₄, V_{i, q}) = d(S₁, V_{i, n + 2 − q}).

These representations are distinct in at least one coordinate. So Π is a resolving partition for M_{m, n} so clearly pd(M_{m, n}) ≤ 4.

Example

Clearly pd(M_9,4) ≤ 4 as the resolving partition for M_9,4 is Π = {S₁, S₂, S₃, S₄} where S₁ = {V_1,1}, S₂ = {V_1,4}, S₃ = {V_1,2, V_1,3, V_2,1, V_2,2, V_2,3, V_2,4,.....

,V_7,1, V_7,2, V_7,4, V_8,2, V_8,3, V_8,4}, S₄ = {V_8,1}.

The representations of different vertices of M_9,4with respect to Π are

V1,1(0,3,1,4),V1,2(1,2,0,3),V1,3(2,1,0,2),V1,4(3,0,1,1),V2,1(1,4,0,5),V2,2(2,3,0,4),V2,3(3,2,0,3),V2,4(4,1,0,2),V3,1(2,5,0,5),V3,2(3,4,0,5),V3,3(4,3,0,4),V3,4(5,2,0,3),V4,1(3,5,0,4),V4,2(4,5,0,5),V4,3(5,4,0,5),V4,4(5,3,0,4),V5,1(4,4,0,3),V5,2(5,5,0,4),V5,3(5,5,0,5),V5,4(4,4,0,5),V6,1(5,3,0,2),V6,2(5,4,0,3),V6,3(4,5,0,4),V6,4(3,5,0,5),V7,1(5,2,0,1),V7,2(4,3,0,2),V7,3(3,4,0,3),V7,4(2,5,0,4),V8,1(4,1,1,0),V8,2(3,2,0,1),V8,3(2,3,0,2),V8,4(1,4,0,3)

Case II: when m and n are of same parity: We want to prove that pd(M_{m, n}) ≤ 5 by constructing a general resolving partition of size 5, for m − n ≥ 4 and m, n are of same parity.

Proof

Let Π = {S₁, S₂, S₃, S₄, S₅} where S₁ = {V_1,1}, S₂ = {V_1,n} , S₃ = {V_1,2, V_1,3,...,V_{1,n − 1}, V_2,1, V_2,2,......,V_2,n,.....,V_{m − 2,1}, V_{m − 2,2},...,V_{m − 2,n}, V_{m − 1,2}, V_{m − 1,3},...,V_{m − 1,n − 1}}, S₄ = {V_{m − 1,1}} , S₅ = {V_{m − 1,n}} .

We prove that Π is a resolving partition for M_{m, n}. To find distance vectors we use two parameters q , i and depending on their different values we divide the entries of distance vectors into five steps.

Step I: Distances of S₁ with all vertices of M_{m, n}.

In this case for each value of q ∈ {1, 2,..., n} the parameter i varies from 1 to m - 1. The entries of different vectors are

d(S1,Vi,q)=i+q−2,1≤i≤12(m+n−2q+2)m+n−q−i,12(m+n−2q+4)≤i≤m−1

Step II: Distances of S₂ with all vertices of M_{m, n}.

For each value of q ∈ {1,2,..., n} the parameter i varies from 1 to m - 1 and we get d(S₂, V_{i, q}) = d(S₁, V_{i, n + 1 − q}).

Step III : Distances of S₃ with all vertices of M_{m, n}.

Here for each value of q ∈ {1,2,..., n} the parameter i varies from 1 to m - 1 and we have

d(S3,Vi,q)=1,ifi=1,q=1,q=n1,ifi=m−1,q=1,q=n0,otherwise

Step IV : Distances of S₄ with all vertices of M_{m, n}. Here we have two parts

a) For q = 1 , we have

d(S4,Vi,q)=i+n−1,1≤i≤12(m−n)m−1−i,12(m−n+2)≤i≤m−1

b) For each value of q ∈ {2, ..., n} the parameter i varies from 1 to m - 1 and we have d(S₄, V_{i, q}) = d(S₁, V_{i, n + 2 − q})

Step V : Distances of S₅ with all vertices of M_{m, n}.

Here for each value of ∈ {1,2,..., n} the parameter i varies from 1 to m - 1 and we have d(S₅, V_{i, q}) = d(S₄, V_{i, n + 1 − q}).

These representations are distinct in at least one coordinate. So Π is a resolving partition for M_{m, n}. Since there is no 4 resolving partition for M_{m, n}, hence Π is a minimal resolving partition for M_{m, n}. So partition dimension of M_{m, n} is 5.

Example

The partition dimension of M_9,3is 5. The resolving partition for M_9,3is Π = {S₁, S₂, S₃, S₄, S₅}. Where

S1={V1,1}S2={V1,3}S3={V1,2,V2,1,V2,2,V2,3,.....,V7,1,V7,2,V7,3,V8,2,V8,3}S4={V8,1}S5={V8,3}

The representations of different vertices of M_9,3 with respect to Π are

V1,1(0,2,1,3,1),V1,2(1,1,0,2,2),V1,3(2,0,1,1,2),V2,1(1,3,0,4,2),V2,2(2,2,0,3,3),V2,3(3,1,0,2,4),V3,1(2,4,0,5,3),V3,2(3,3,0,4,4),V3,3(4,2,0,3,5),V4,1(3,5,0,4,4),V4,2(4,4,0,5,5),V4,3(5,3,0,4,4),V5,1(4,4,0,3,5),V5,2(5,5,0,4,4),V5,3(4,4,0,5,3),V6,1(5,3,0,2,4),V6,2(4,4,0,3,3),V6,3(3,5,0,4,2),V7,1(4,2,0,1,3),V7,2(3,3,0,2,2),V7,3(2,4,0,3,1),V8,1(3,1,1,0,2),V8,2(2,2,0,1,1),V8,3(1,3,1,2,0,)

3.2 Lower bound

Proof

It is clear that 2 < pd(M_{m, n}) as it is not a path, [8]. So it is obvious that 3 ≤ pd(M_{n, m}). □

Theorem 3.2.

For m ≥ 3 and n ≥ 2

2≤β(Mm,n)≤4,whenn≡1(mod2)andm≡1(mod2),m−n≥43,whenn≡0(mod2)andm≡1(mod2)3whenn≡1(mod2)andm≡0(mod2)4whenn≡0(mod2)andm≡0(mod2),m−n≥4

Proof

Proof is just straightforward after taking into account the fundamental inequality between metric and patrtition dimensions. □

4 Conclusions and open problems

In this article we have computed sharp upper bounds for the partition dimension of the generalized Möbius ladders and arrive at the following results

Theorem 4.1.

For m ≥ 3 and n ≥ 2

3≤pd(Mm,n)≤5,whenn≡1(mod2)andm≡1(mod2),m−n≥44,whenn≡0(mod2)andm≡1(mod2)4whenn≡1(mod2)andm≡0(mod2)5whenn≡0(mod2)andm≡0(mod2),m−n≥4

and

Theorem 4.2.

For m ≥ 3 and n ≥ 2

2≤β(Mm,n)≤4,whenn≡1(mod2)andm≡1(mod2),m−n≥43,whenn≡0(mod2)andm≡1(mod2)3whenn≡1(mod2)andm≡0(mod2)4whenn≡0(mod2)andm≡0(mod2),m−n≥4

At the same time we pose natural open problems regarding the exact values of partition dimension, pd(M_{m, n}) and β(M_{m, n}), and sharp lower bounds for this new family of graphs. For further problems about the dimensions of graphs please see [28, 29].

Competing interests The authors declare that they have no competing interests.
Author’s contributions All authors contributed equally to the writing of this paper. All authors read and approved the final manuscript.

References

[1] Harary F., Melter R. A., On the metric dimension of a graph, Ars. Combinatoria, 1976, 2, 191-195.Search in Google Scholar

[2] Sebo A., Tannier E., On metric generators of graphs, Math. Oper. Res., 2004, 29, 383-393.10.1287/moor.1030.0070Search in Google Scholar

[3] Bogomonly A., and Greenwell D., Cut the knote: Invitition to Mastermind, 1999. http://www.maa.org/editorial/Knot/Mastermind.html.Search in Google Scholar

[4] Chvatal V., Mastermind, Combinatorica, 1983, 3, 325-329.10.1007/BF02579188Search in Google Scholar

[5] Khuller S., Raghavachari B., Rosenfeld A., Landmarks in graphs, Disc. Appl. Math. 1996, 70, 217-229.10.1016/0166-218X(95)00106-2Search in Google Scholar

[6] Frank P., Silverman R., Remarks on detection problems, Amer. Math. Monthly, 1967, 74, 171-173.10.2307/2315611Search in Google Scholar

[7] Melter R.A., Tomescu I., Metric bases in digital geometry, Computer Vision, Graphics, and Image Processing, 1984, 25, 113-121.10.1016/0734-189X(84)90051-3Search in Google Scholar

[8] Chartrand G., Eroh L., Johnson M. A., Oellermann, O. R.,Resolvibility in graphs and the metric dimension of a grap, Disc. Appl. Math., 2000, 105, 99-133.10.1016/S0166-218X(00)00198-0Search in Google Scholar

[9] Buczkowski P.S., Chartrand G., Poisson C., Zhang, P. On k-dimensional graphs and their bases, Pariodica Math. Hung, 2003, 46, 9-15.10.1023/A:1025745406160Search in Google Scholar

[10] Caceres J., Hernando C., Mora M., Pelayo I.M., Puertas M.L., Seara C., Wood D.R., On the metric dimension of some families of graphs, Electronic Notes in Disc. Math., 2005, 22, 129-133.10.1016/j.endm.2005.06.023Search in Google Scholar

[11] Tomescu I., Javaid I., On the metric dimension of the Jahangir graph, Bull. Math. Soc. Sci. Math. Roumanie, 2007, 50, 371-376.Search in Google Scholar

[12] Javaid I., Rahim M.T., Ali K., Families of regular graphs with constant metric dimension, Utilitas Math., 2008, 75, 21-33.Search in Google Scholar

[13] Shanmukha B., Sooryanarayana B., Harinath K. S., Metric dimension of wheels, Far East J. Appl. Math. 2002, 8, 217-229.Search in Google Scholar

[14] Poisson C., Zhang P., The metric dimension of unicyclic graphs, J. Comb. Math Comb. Comput. 2002, 40, 17-32.Search in Google Scholar

[15] Hussain Z., Munir M., Chaudhary M., Kang S.M. Computing Metric Dimension and Metric Basis of 2D Lattice of Alpha-Boron Nanotubes, Symmetry 2018, 10, 300.10.3390/sym10080300Search in Google Scholar

[16] Murtaza A., Ali G., Imaran M., Baig A.Q., Kashif M., On the metric dimension of Möbius Ladder, ARS Combinatoria, 2012 105, 403-410.Search in Google Scholar

[17] Munir M., Nizami A. R., Saeed H., Iqba Z., On the metric dimension of Möbius Ladder, ARS Combinatoria, 2017, 135, 239-245.Search in Google Scholar

[18] Chartrand G., Poisson C., Zhang P., Resolvability and the upper dimension of graphs. Comput. Math. Appl., 2000, 39, 19-28.10.1016/S0898-1221(00)00126-7Search in Google Scholar

[19] Slater P. J., Dominating and refrences sets in graphs, J. Math. Phys. sci., 1998, 22, 445-455.Search in Google Scholar

[20] Chartrand G., Salehi E., Zhang P., On the partition dimension of a graph, Congr. Numer., 1998, 131, 55-66.10.1007/PL00000127Search in Google Scholar

[21] Chartrand G., Salehi E., Zhang P., The partition dimension of a graph, Aequationes Math, 2000, 59, 45-54.10.1007/PL00000127Search in Google Scholar

[22] Tomescu I., Imran M., On metric and partition dimensions of some infinite regular graphs, Bull. Math. Soc. Sci. Math. Roumanie 2009, 100, 461-472.Search in Google Scholar

[23] Tomescu I., Imran M., Slamin M., On the partition dimension and connected partition dimension of wheels, Ars Combinatoria, 2007, 84, 311-317.Search in Google Scholar

[24] Chappell C., Glenn G., Gimbel J. Hartman C., Bounds on the metric and partition dimensions of a graph, Ars Combinatoria, 2008, 88, 349-366.Search in Google Scholar

[25] Fernau H., Rodríguez-Velázquez J. A., Yero I. G., On the partition dimension of unicyclic graphs, Bull. Math. Soc. Sci. Math. Roumanie, Tome 2014, 57, 381-391.Search in Google Scholar

[26] Rojas A., Diaz K., Distance Labellings of Möbius Ladders, disertaion Worcester Polytechnic Institute 12-3-2013.Search in Google Scholar

[27] Hongbin M., Idrees M., Nizami A.R., Munir M., Generalized Möbius Ladder and Its Metric Dimension, arXiv:1708.05199.Search in Google Scholar

[28] Tang Z., Liang L., Gao W., Wiener polarity index of quasi-tree molecular structures, Open J. Math. Sci., 2018, 1, 73-83.10.30538/oms2018.0018Search in Google Scholar

[29] Umar M.A., Javed M.A., Hussain M., Ali B.R., Super (a, d) - C 4 -antimagicness of book graphs Open J. Math. Sci., 2018, 2, 115-121.10.30538/oms2018.0021Search in Google Scholar

Received: 2018-03-24

Accepted: 2018-10-05

Published Online: 2018-11-08

This work is licensed under the Creative Commons Attribution-NonCommercial-NoDerivatives 4.0 License.

Articles in the same Issue

https://doi.org/10.1515/math-2018-0109

Keywords for this article

Metric dimension; Partition dimension; Generalized Möbius Ladder

Creative Commons

BY-NC-ND 4.0