TARU PUBLICATIONS
Journal of Discrete Mathematical Sciences and Cryptography cover

Journal of Discrete Mathematical Sciences and Cryptography

(Since 1998)
Open Access ·Peer-reviewed·ISSN (Online): 2169-0065·ISSN (Print): 0972-0529

CiteScore 2025: 2.7 (Q1)

Powered by:Powered by

Monthly Journal: Publishes theoretical and applied research in all areas of Discrete Mathematical Sciences, Cryptography, Combinatorics, Elliptic Curves and Information Security.

Issues up to 2022 co-published with and available at:Taylor & Francis Online
submissions@tarupublications.com
Taru Online First Current IssueSpecial IssuesBrowse all Issues
Journal of Discrete Mathematical Sciences and Cryptography
Open Access Research Article

Dominant metric dimension of rooted product graph

, * , , ,

* Corresponding author · click or hover a name for details

pp. 2409–2420Vol. 29Issue 6June 2026DOI: 10.47974/JDMSC-2656 Crossmark XML
Received:
01 Aug 2025
Published Online:
13 Jun 2026
Article type:
Research Article
Language:
EN
Article no.:
JDMSC-2656
Pages:
2409–2420

Abstract

Poverty is closely related to the problem of unequal distribution of basic needs across regions. To address this challenge, graph theory provides a rigorous framework for designing efficient distribution networks that ensure accessibility while minimizing cost. In this work, we examine the dominant metric dimension, explained as the smallest possible size of a set that is both resolving and dominating. The primary finding shows that for the graph formed via the rooted product G ∘ 𝓗, this dimension aligns to the sum of the dominant metric dimensions of all graphs within the family 𝓗. When the sequence consists of complete graph, star graph, and complete bipartite graph with the rooted vertex at a pendant, the result should be reduced by n. These findings provide insight into how the number of distribution centers can be optimized in various network structures. From the perspective of poverty alleviation, this result implies that efficient placement of distribution hubs can reduce infrastructure costs while ensuring that every community is within reach of essential goods. This paper determines the dominant metric dimension of the specified product G ∘ 𝓗, represented as Ddim(G ∘ 𝓗). The findings indicate that the value of Ddim(G ∘ 𝓗) depends on the structural properties of the graphs within the sequence 𝓗. When 𝓗 consists of rooted cycle graphs, rooted star graphs, and rooted path graphs, the dominant metric dimension of G ∘ 𝓗 is equal to the total sum of the dominant metric dimensions of each element in 𝓗. On the other hand, when H consists of rooted complete graph, Ddim(G ∘ 𝓗) is the sum of their individual dominant metric dimensions, each reduced by one.

Keywords

Alleviate povertyDominant setDominant resolving setResolving setRooted product graph

Subject Classifications

05C1205C7505C38

References

[1] G. Chartrand, L. Eroh, M.A. Johnson, and O.R. Oellermann, R”Resolv­ability in graphs and the metric dimension of a graph,” Discrete Appl. Math., vol. 105, pp. 99–113 (2000).
[2] F. Harary and R. A. Melter, “On the metric dimension of a graph,” Ars Combinatoria, vol. 2, pp. 191–195 (1976).
[3] J. Bensmail, F. Mc Inerney, and N. Nisse, “Metric dimension: from graphs to oriented graphs,” Electronic Notes in Theoretical Computer Science, vol. 346, pp. 111–123 (2019).
[4] F. Okamoto, L. Crosse, B. Phinezy, P. Zhang, and Kalamazoo, “The local metric dimension of a graph,” Mathematica Bohemica, vol. 135, no. 3, pp. 239–255 (2010).
[5] M. Jannesari and B. Omoomi, “The metric dimension of the lexicographic product of graphs,” Discrete Mathematics, vol. 312, pp. 3349–3356 (2012).
[6] I. G. Yero, D. Kuziak, and J. A. Rodríguez-Velázquez, “On the metric dimension of corona product graphs,” Computers & Mathematics with Applications, Elsevier, vol. 61, no. 9, pp. 2793–2798 (2011).
[7] S. W. Saputro, D. Suprijanto, E. T. Baskoro, and A. N. M. Salman, “The metric dimension of a graph composition product with star,” Journal of the Indonesian Mathematical Society, vol. 18, pp. 85–92 (2012).
[8] S. W. Saputro, N. Mardiana, and I. A. Purwasih, “The metric dimension of comb product graphs,” Matematicki Vesnik, vol. 69, no. 4, pp. 248–258 (2017).
[9] H. Iswadi, E. T. Baskoro, A. Salman, and R. Simanjuntak, “The metric dimension of amalgamation cycle,” Far East Journal of Mathematical Sciences, vol. 41, no. 1, pp. 19–31 (2010).
[10] L. Susilowati, I. Sa’adah, R. Z. Fauziyyah, A. Erfanian, and Slamin, “The dominant metric dimension of graphs,” Heliyon, vol. 6, no. 3, p. e03633 (2020).
[11] M. Monica and S. Santhakumar, “Partition dimension of rooted product graphs,” Discrete Applied Mathematics, vol. 262, pp. 138–147 (2019).
[12] M. Sardar, M. Alaeiyan, M. Farahani, M. Cancan, and S. Ediz, “Resistance distance in some classes of rooted product graphs obtained by Laplacian Generalized” Inverse Method, Journal of Information and Optimization Sciences, vol. 42, pp. 1–21 (2021).
[13] P. Jeyanthi, M. Selvi, and D. Ramya, “Restricted triangular difference mean graphs,” Journal of Mathematics, vol. 39, no. 2, pp. 275–286 (2020).
[14] C. D. Godsil and B. D. McKay, “A new graph and its spectrum,” Bulletin of the Australian Mathematical Society, vol. 18, pp. 21–28 (1978).
[15] T. W. Damayanti, L. Susilowati, N. Estuningsih, M. Y. Syaifuddin, and Bustomi, “The combination concept of locating and dominating set of rooted product graphs,” AIP Conference Proceedings, vol. 3326, 030007 (2026).

Views: 44Downloads: 16Citations: 0