On the dominating partition dimension of Möbius ladders and burger graphs
Abstract
Let G be a connected graph and let Π = {S1, S2, …, Sk} be a partition of V(G). The partition Π is called a dominating resolving partition of G if distinct vertices have distinct distance representations with respect to Π, and every vertex has a neighbor in a partition class different from its own. The minimum number of classes in such a partition is called the dominating partition dimension of G, denoted by ηp(G). In this paper, we determine the dominating partition dimension of two families of cubic graphs: Möbius ladders and burger graphs with range one. We show that the dominating partition dimension is equal to 4 for both families within the specified ranges.
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PDFDOI: http://dx.doi.org/10.5614/ejgta.2026.14.2.13
References
A. Asmiati, A. A. Rachmawati, D. E. Nurvazly, and N. Hamzah, The partition dimension of daisy graphs and its barbell, Science and Technology Indonesia 10 (2025), 313–319.
E. T. Baskoro and D. O. Haryeni, All graphs of order n ≥ 11 and diameter 2 with partition dimension n − 3, Heliyon 6 (2020), e03694.
R. Bhatti, M. K. Jamil, M. Azeem, and P. Poojary, Partition dimension of generalized hexagonal cellular networks and its application, IEEE Access 12 (2024), 12199–12208.
R. C. Brigham, G. Chartrand, R. D. Dutton, and P. Zhang, Resolving domination in graphs, Math. Bohem. 128 (2003), 25–36.
G. Chartrand, D. Erwin, M. A. Henning, P. J. Slater, and P. Zhang, The locating-chromatic number of a graph, Bull. Inst. Combin. Appl. 36 (2002), 101–110.
G. Chartrand, L. Eroh, M. A. Johnson, and O. R. Oellermann, Resolvability in graphs and the metric dimension of a graph, Discrete Appl. Math. 105 (2000), 99–113.
G. Chartrand, E. Salehi, and P. Zhang, On the partition dimension of a graph, Congr. Numer. 130 (1998), 157–160.
G. Chartrand, E. Salehi, and P. Zhang, The partition dimension of a graph, Aequationes Math. 59 (2000), 45–54.
G. Chartrand and P. Zhang, The theory and applications of resolvability in graphs: A survey, Congr. Numer. 160 (2003), 47–68.
Y. Hafidh and E. T. Baskoro, Partition dimension of trees—palm approach, Electron. J. Graph Theory Appl. 12 (2) (2024), 265–272.
F. Harary and R. A. Melter, On the metric dimension of a graph, Ars Combin. 2 (1976), 191–195.
M. A. Henning and O. R. Oellermann, Metric-locating-dominating sets in graphs, Ars Combin. 73 (2004), 129–141.
C. Hernando, M. Mora, and I. M. Pelayo, Resolving dominating partitions in graphs, Discrete Appl. Math. 266 (2019), 237–251.
Z. Hussain, J. A. Khan, M. Munir, M. S. Saleem, and Z. Iqbal, Sharp bounds for partition dimension of generalized Möbius ladders, Open Math. 16 (2018), 1283–1290.
S. Khuller, B. Raghavachari, and A. Rosenfeld, Landmarks in graphs, Discrete Appl. Math. 70 (1996), 217–229.
O. Ore, Theory of Graphs, American Mathematical Society, Providence, RI, 1962.
R. N. Raj, R. S. Rajan, and H. Ahmad, A comprehensive analysis of partition dimensions in Efavirenz Abacavir Lamivudine Doravirine of anti-HIV drug structures, Iraqi Journal for Computer Science and Mathematics 5 (2024), 100–107.
M. Ridwan, H. Assiyatun, and E. T. Baskoro, The dominating partition dimension and locating-chromatic number of graphs, Electron. J. Graph Theory Appl. 11 (2) (2023), 455–465.
M. Ridwan and E. T. Baskoro, On graphs of order m with dominating partition dimension m − 3, AIP Conference Proceedings 2423 (2021), 060010.
M. Ridwan and W. Abidin, On trees with locating-chromatic number and dominating partition dimension equal to three, Jurnal Matematika dan Statistika serta Aplikasinya 13(2) (2025), 220–223.
P. J. Slater, Leaves of trees, Congr. Numer. 14 (1975), 549–559.
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