For a simple connected graph $ G = (V, E) $, a vertex $ x\in V $ distinguishes two elements (vertices or edges) $ x_1\in V, y_1 \in E $ if $ d(x, x_1)\neq d(x, y_1). $ A subset $ Q_m\subset V $ is a mixed metric generator for $ G, $ if every two distinct elements (vertices or edges) of $ G $ are distinguished by some vertex of $ Q_m. $ The minimum cardinality of a mixed metric generator for $ G $ is called the mixed metric dimension and denoted by $ dim_m(G). $ In this paper, we investigate the mixed metric dimension for different families of ladder networks. Among these families, we consider Möbius ladder, hexagonal Möbius ladder, triangular Möbius ladder network and conclude that all these families have constant-metric, edge metric and mixed metric dimension.
Citation: Mohra Zayed, Ali Ahmad, Muhammad Faisal Nadeem, Muhammad Azeem. The comparative study of resolving parameters for a family of ladder networks[J]. AIMS Mathematics, 2022, 7(9): 16569-16589. doi: 10.3934/math.2022908
For a simple connected graph $ G = (V, E) $, a vertex $ x\in V $ distinguishes two elements (vertices or edges) $ x_1\in V, y_1 \in E $ if $ d(x, x_1)\neq d(x, y_1). $ A subset $ Q_m\subset V $ is a mixed metric generator for $ G, $ if every two distinct elements (vertices or edges) of $ G $ are distinguished by some vertex of $ Q_m. $ The minimum cardinality of a mixed metric generator for $ G $ is called the mixed metric dimension and denoted by $ dim_m(G). $ In this paper, we investigate the mixed metric dimension for different families of ladder networks. Among these families, we consider Möbius ladder, hexagonal Möbius ladder, triangular Möbius ladder network and conclude that all these families have constant-metric, edge metric and mixed metric dimension.
| [1] | P. Slater, Leaves of trees, Proceedings of the 6th Southeastern Conference on Combinatorics, Graph Theory, and Computing, Congressus Numerantium, 14 (1975), 549–559. |
| [2] | F. Harary, R. Melter, On the metric dimension of a graph, Ars Combinatoria, 2 (1976), 191–195. |
| [3] |
A. Kelenc, N. Tratnik, I. Yero, Uniquely identifying the edges of a graph: the edge metric dimension, Discrete Appl. Math., 251 (2018), 204–220. http://dx.doi.org/10.1016/j.dam.2018.05.052 doi: 10.1016/j.dam.2018.05.052
|
| [4] |
A. Kelenc, D. Kuziak, A. Taranenko, I. Yero, Mixed metric dimension of graphs, Appl. Math. Comput., 314 (2017), 429–438. http://dx.doi.org/10.1016/j.amc.2017.07.027 doi: 10.1016/j.amc.2017.07.027
|
| [5] |
G. Chartrand, L. Eroh, M. Johnson, O. Ortrud, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math., 105 (2000), 99–113. http://dx.doi.org/10.1016/S0166-218X(00)00198-0 doi: 10.1016/S0166-218X(00)00198-0
|
| [6] |
S. Khuller, B. Raghavachari, A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math., 70 (1996), 217–229. http://dx.doi.org/10.1016/0166-218X(95)00106-2 doi: 10.1016/0166-218X(95)00106-2
|
| [7] |
V. Chvátal, Mastermind, Combinatorica, 3 (1983), 325–329. http://dx.doi.org/10.1007/BF02579188 doi: 10.1007/BF02579188
|
| [8] | P. Erdös, A. Rényi, On two problems of information theory, Magyar Tud. Akad. Mat. Kutató Int. Közl, 8 (1963), 229–243. |
| [9] | B. Lindström, On a combinatory detection problem, Magyar Tud. Akad. Mat. Kutató Int. Közl, 9 (1964), 195–207. |
| [10] | E. Badr, K. Aloufi, A robot's response acceleration using the metric dimension problem, submitted for publication. http://dx.doi.org/10.20944/preprints201911.0194.v1 |
| [11] |
B. Deng, M. Nadeem, M. Azeem, On the edge metric dimension of different families of Möbius networks, Math. Probl. Eng., 2021 (2021), 6623208. http://dx.doi.org/10.1155/2021/6623208 doi: 10.1155/2021/6623208
|
| [12] | M. Ali, G. Ali, M. Imran, A. Baig, M. Shafiq, On the metric dimension of Möbius ladders, Ars Combinatoria, 105 (2012), 403–410. |
| [13] |
M. Nadeem, M. Azeem, A. Khalil, The locating number of hexagonal Möbius ladder network, J. Appl. Math. Comput., 66 (2021), 149–165. http://dx.doi.org/10.1007/s12190-020-01430-8 doi: 10.1007/s12190-020-01430-8
|
| [14] |
D. Kuziak, J. Rodríguez-Velázquez, I. Yero, On the strong metric dimension of product graphs, Electronic Notes in Discrete Mathematics, 46 (2014), 169–176. http://dx.doi.org/10.1016/j.endm.2014.08.023 doi: 10.1016/j.endm.2014.08.023
|
| [15] |
H. Alshehri, A. Ahmad, Y. Alqahtani, M. Azeem, Vertex metric-based dimension of generalized perimantanes diamondoid structure, IEEE Access, 10 (2022), 43320–43326. http://dx.doi.org/10.1109/ACCESS.2022.3169277 doi: 10.1109/ACCESS.2022.3169277
|
| [16] |
A. Koam, A. Ahmad, M. Azeem, A. Khalil, M. Nadeem, On adjacency metric dimension of some families of graph, J. Funct. Space., 2022 (2022), 6906316. http://dx.doi.org/10.1155/2022/6906316 doi: 10.1155/2022/6906316
|
| [17] |
A. Koam, A. Ahmad, M. Azeem, M. Nadeem, Bounds on the partition dimension of one pentagonal carbon nanocone structure, Arab. J. Chem., 15 (2022), 103923. http://dx.doi.org/10.1016/j.arabjc.2022.103923 doi: 10.1016/j.arabjc.2022.103923
|
| [18] |
M. Azeem, M. Imran, M. Nadeem, Sharp bounds on partition dimension of hexagonal Möbius ladder, J. King Saud Univ. Sci., 34 (2022), 101779. http://dx.doi.org/10.1016/j.jksus.2021.101779 doi: 10.1016/j.jksus.2021.101779
|
| [19] |
M. Azeem, M. Nadeem, Metric-based resolvability of polycyclic aromatic hydrocarbons, Eur. Phys. J. Plus, 136 (2021), 395. http://dx.doi.org/10.1140/epjp/s13360-021-01399-8 doi: 10.1140/epjp/s13360-021-01399-8
|
| [20] |
M. Ali, G. Ali, U. Ali, M. Rahim, On cycle related graphs with constant metric dimension, Open Journal of Discrete Mathematics, 2 (2012), 21–23. http://dx.doi.org/10.4236/ojdm.2012.21005 doi: 10.4236/ojdm.2012.21005
|
| [21] |
R. Adawiyah, D. Dafik, R. Alfarisi1, R. Prihandini, I. Agustin, M. Venkatachalam, The local edge metric dimension of graph, J. Phys.-Conf. Ser., 1543 (2020), 012009. http://dx.doi.org/10.1088/1742-6596/1543/1/012009 doi: 10.1088/1742-6596/1543/1/012009
|
| [22] |
Y. Zhang, S. Gao, On the edge metric dimension of convex polytopes and its related graphs, J. Comb. Optim., 39 (2020), 334–350. http://dx.doi.org/10.1007/s10878-019-00472-4 doi: 10.1007/s10878-019-00472-4
|
| [23] |
H. Raza, Y. Ji, Computing the mixed metric dimension of a generalized Petersen graph $P(n, 2)$, Front. Phys., 8 (2020), 211. http://dx.doi.org/10.3389/fphy.2020.00211 doi: 10.3389/fphy.2020.00211
|
| [24] |
H. Raza, J. Liu, S. Qu, On mixed metric dimension of rotationally symmetric graphs, IEEE Access, 8 (2020), 11560–11569. http://dx.doi.org/10.1109/ACCESS.2019.2961191 doi: 10.1109/ACCESS.2019.2961191
|
| [25] |
A. Ahmad, M. Ba$\breve {\rm{c}}$a, S. Sultan, Computing the metric dimension of kayak paddles graph and cycles with chord, Proyecciones, 39 (2020), 287–300. http://dx.doi.org/10.22199/issn.0717-6279-2020-02-0018 doi: 10.22199/issn.0717-6279-2020-02-0018
|
| [26] |
J. Liu, M. Nadeem, H. Siddiqui, W. Nazir, Computing metric dimension of certain families of Toeplitz graphs, IEEE Access, 7 (2019), 126734–126741. http://dx.doi.org/10.1109/ACCESS.2019.2938579 doi: 10.1109/ACCESS.2019.2938579
|
| [27] |
J. Liu, A. Zafari, H. Zarei, Metric dimension, minimal doubly resolving sets, and the strong metric dimension for jellyfish graph and cocktail party graph, Complexity, 2020 (2020), 9407456. http://dx.doi.org/10.1155/2020/9407456 doi: 10.1155/2020/9407456
|
| [28] |
J. Liu, Z. Zahid, R. Nasir, W. Nazeer, Edge version of metric dimension and doubly resolving sets of the necklace graph, Mathematics, 6 (2018), 243. http://dx.doi.org/10.3390/math6110243 doi: 10.3390/math6110243
|
| [29] |
I. Yero, Vertices, edges, distances and metric dimension in graphs, Electronic Notes in Discrete Mathematics, 55 (2016), 191–194. http://dx.doi.org/10.1016/j.endm.2016.10.047 doi: 10.1016/j.endm.2016.10.047
|
Figures(5) / Tables(1)
Mohra Zayed, Ali Ahmad, Muhammad Faisal Nadeem, Muhammad Azeem. The comparative study of resolving parameters for a family of ladder networks[J]. AIMS Mathematics, 2022, 7(9): 16569-16589. doi: 10.3934/math.2022908
DownLoad: