ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS

Hamdy Hafez

ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS

Abstract

Recently, rough graphs have received considerable attention for their ability to model imprecise information in graphical data. By leveraging rough set theory, rough graphs address uncertainty through approximation and classification mechanisms. In this paper, we introduce rough graphs as mathematical objects defined through graph automorphism groups, providing a symmetry-based framework for uncertainty. Concurrently, we derive an indiscernibility relation from binary relations $R$ via the automorphism group of its graph representation $G(R)$, where indiscernibility corresponds to orbits under the group action. This approach generalizes to define rough relations and rough graphs through an arbitrary permutation group acting on the underlying set.

Keywords

References

[1] Bondy, J. A., Murty, U. S., (2008), Graph Theory, Springer London.
[2] Cao, L., Huang, G., (2017), Concept design and construction algorithm of rough complex networks, J. Intell. & Fuzzy Syst., 33 (3), pp. 1441-1451.
[3] Cao, L., Huang, G., Chai, W., (2017), A knowledge discovery model for third-party payment networks based on rough set theory, J. Intell. & Fuzzy Syst., 33 (1), pp. 413-421.
[4] Chen, J., Li, J., (2012), An application of rough sets to graph theory, Inf. Sci., 201, pp. 114-127.
[5] Chen, J., Mi, J., Lin, Y., (2020), A graph approach for fuzzy-rough feature selection, Fuzzy Sets Syst., 391, pp. 96-116.
[6] Chiaselotti, G., Ciucci, D., Gentile, T., (2015), Simple undirected graphs as formal contexts, In: International Conference on Formal Concept Analysis, pp. 287-302, Springer.
[7] Chiaselotti, G., Ciucci, D., Gentile, T., Infusino, F., (2015), Rough set theory applied to simple undirected graphs, In: International Conference on Rough Sets and Knowledge Technology, pp. 423- 434, Springer.
[8] El Atik, A., Nawar, A., Atef, M., (2021), Rough approximation models via graphs based on neighborhood systems, Granul. Comput., 6, pp. 1025-1035.

[9] Fomin, F. V., Golovach, P. A., Strømme, T. J. F., Thilikos, D. M., (2020), Subgraph complementation, Algorithmica, 82 (7), pp. 1859-1880.
[10] Fuentes, I., Pina, A., Napoles, G., Arco, L., Vanhoof, K., (2020), Rough net approach for community detection analysis in complex networks, In: Rough Sets: International Joint Conference, IJCRS 2020, pp. 401-415, Springer.
[11] Ge, X., Li, Z., (2011), Definable subsets in covering approximation spaces, Int. J. Math. Comput. Sci., 5 (3), pp. 475-478.
[12] Harary, F., (1970), Graph Theory, Addison-Wesley.
[13] Harary, F., Read, R. C., (2006), Is the null-graph a pointless concept?, In: Graphs and Combinatorics, Proceedings of the Capital Conference on Graph Theory and Combinatorics at the George Washington University June 18-22, pp. 37-44, Springer.
[14] He, T., Shi, K., (2006), Rough graph and its structure, J. Shandong Univ., 41 (6), pp. 46-50.
[15] Kaminski, M., Lozin, V. V., Milanic, M., (2009), Recent developments on graphs of bounded clique-width, Discrete Appl. Math., 157 (12), pp. 2747-2761.
[16] Liang, M., Liang, B., Wei, L., Xu, X., (2011), Edge rough graph and its application,In: 2011 Eighth International Conference on Fuzzy Systems and Knowledge Discovery (FSKD), 1, pp. 335-338, IEEE.
[17] Mathew, B., John, S. J., Garg, H., (2020), Vertex rough graphs, Complex Intell. Syst., 6 (2), pp. 347-353.
[18] Pawlak, Z., (1982), Rough sets, Int. J. Comput. Inf. Sci., 11, pp. 341-356.
[19] Pawlak, Z., (1986), Rough relations, Bull. Pol. Acad. Sci. Tech. Sci., 34 (9-10), pp. 557-590.
[20] Pei, D., (2007), On definable concepts of rough set models, Inf. Sci., 177 (19), pp. 4230-4239.
[21] Peters, J. F., Skowron, A., (2002), A rough set approach to knowledge discovery, Int. J. Intell. Syst., 17 (2), pp. 109-112.
[22] Polkowski, L., (2013), Rough sets in knowledge discovery 2: applications, case studies and software systems, Physica, 19.
[23] Qiao, S., Zhu, P., Pedrycz, W., (2022), Rough set analysis of graphs, Filomat, 36 (10), pp. 3331-3354.
[24] Sarwar, M., (2020), Decision-making approaches based on color spectrum and d-topsis method under rough environment, Comput. Appl. Math., 39, p. 291.
[25] Skowron, A., Nguyen, H. S., Jankowski, A., (2017), Rough sets in pattern recognition, In: Pattern Recognit. Big Data, pp. 323-393, World Scientific.
[26] Wang, D., Zhu, P., (2022), Graph reduction in a path information-based rough directed graph model, Soft Comput., 26 (9), pp. 4171-4186.
[27] Yao, Y. Y., (1996), Two views of the theory of rough sets in finite universes, Int. J. Approx. Reason., 15 (4), pp. 291-317.
[28] Zakowski, W., (1983), Approximations in the space (u, π), Demonstr. Math., 16 (3), pp. 761-770.
[29] Javaid, I., Ali, S., Ur Rehman, S., Shah, A., (2022), Rough sets in graphs using similarity relations, AIMS Math., 7 (4), pp. 5790-5807.
[30] Javaid, I., Fatima, A., Akram, M., (2025), Symmetry-based granulation in networks associated with commutative rings: application in social network dynamics, J. Appl. Math. Comput., pp. 1-26.
[31] Shahzamanian, M. H., Shirmohammadi, M., Davvaz, B., (2010), Roughness in cayley graphs, Inf. Sci., 180 (17), pp. 3362-3372.
[32] Biggs, N., White, A. T., (1979), Permutation groups and combinatorial structures, Cambridge University Press, 33.
[33] Fan, T. F., (2013), Rough set analysis of relational structures, Inf. Sci., 221, pp. 230-244.
[34] Lauri, J., Scapellato, R., (2016), Topics in graph automorphisms and reconstruction, Cambridge University Press, 432.
[35] Shaheen, T., Stell, J. G., (2019), Graphical partitions and graphical relations, Fundam. Inform., 165 (1), pp. 75-98.

Details

Primary Language

English

Subjects

Combinatorics and Discrete Mathematics (Excl. Physical Combinatorics)

Journal Section

Research Article

Authors

Hamdy Hafez ^*
0000-0001-6013-8392
Egypt

Publication Date

October 6, 2026

Submission Date

July 11, 2025

Acceptance Date

November 13, 2025

Published in Issue

Year 2026 Volume: 16 Number: 10

IZ

https://izlik.org/JA35ZJ54WP

Cite

RIS / Bibtex

APA

Hafez, H. (2026). ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS. TWMS Journal of Applied and Engineering Mathematics, 16(10), 1207-1221. https://izlik.org/JA35ZJ54WP

AMA

1.Hafez H. ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS. JAEM. 2026;16(10):1207-1221. https://izlik.org/JA35ZJ54WP

Chicago

Hafez, Hamdy. 2026. “ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS”. TWMS Journal of Applied and Engineering Mathematics 16 (10): 1207-21. https://izlik.org/JA35ZJ54WP.

EndNote

Hafez H (October 1, 2026) ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS. TWMS Journal of Applied and Engineering Mathematics 16 10 1207–1221.

IEEE

[1]H. Hafez, “ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS”, JAEM, vol. 16, no. 10, pp. 1207–1221, Oct. 2026, [Online]. Available: https://izlik.org/JA35ZJ54WP

ISNAD

Hafez, Hamdy. “ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS”. TWMS Journal of Applied and Engineering Mathematics 16/10 (October 1, 2026): 1207-1221. https://izlik.org/JA35ZJ54WP.

JAMA

1.Hafez H. ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS. JAEM. 2026;16:1207–1221.

MLA

Hafez, Hamdy. “ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS”. TWMS Journal of Applied and Engineering Mathematics, vol. 16, no. 10, Oct. 2026, pp. 1207-21, https://izlik.org/JA35ZJ54WP.

Vancouver

1.Hamdy Hafez. ROUGH GRAPH APPROXIMATIONS VIA SYMMETRY GROUPS: AUTOMORPHISMS AND PERMUTATIONS. JAEM [Internet]. 2026 Oct. 1;16(10):1207-21. Available from: https://izlik.org/JA35ZJ54WP