Soft Graph On An Equivalence Class Of A Minimal Dominating Set Of A Graph

Supriya M D; P. Usha

doi:10.29284/1s5vmz36

Authors

Supriya M D Department of Mathematics, KSIT (Affiliated to Visvesvaraya Technological University, Belagavi), Bengaluru 560109, India.
P. Usha Department of Mathematics, Siddaganga Institute of Technology (Affiliated to Visvesvaraya Technological University, Belagavi), Tumakuru 572103, India.

DOI:

https://doi.org/10.29284/1s5vmz36

Keywords:

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Abstract

In this paper, we explore a novel soft graph structure derived from equivalence class of a minimal dominating set in a simple graph. We define a soft graph where is an undirected graph and is a minimal dominating set. A mapping is defined as and a mapping is defined as } where and is a power set. The pair is a soft set over and is a soft set over . Then is a subgraph of .

This framework enables decentralized control, modular subgraph-based planning, dynamic adaptability, and reliable routing, making it well suited for emergency management. The proposed model further supports adaptive reallocation strategies and localized optimization, and can be integrated into AI-driven disaster response systems. Its real-world applicability is illustrated through simulations of relief deployment and coverage in a hypothetical urban network.

References

[1] D. Molodtsov, Soft set theory—first results, Computers & Mathematics with Applications 37 (1999) 19–31. doi:10.1016/S0898-1221(99)00056-5.

[2] P. Maji, R. Biswas, A. Roy, Soft set theory, Computers & Mathematics with Applications 45 (4) (2003) 555–562. doi:https: 230 //doi.org/10.1016/S0898-1221(03)00016-6. URL https://www.sciencedirect.com/science/article/pii/ S0898122103000166 11.

[3] M. Gillbert rani, A. Jeyaseelan, Domination in soft graphs, International Journal of Mathematical Archive 6 (3) (2015) 224–226. 235.

[4] R. K. Thumbakara, B. George, Soft graphs, ICSRS Publication 21 (2) (2014) 75–86.

[5] K. Palani, T. Jones, V. Maheswari, Soft graphs of certain graphs, Journal of Physics: Conference Series 1947 (2021) 012045. doi:10.1088/1742-6596/ 1947/1/012045. 240.

[6] M. Akram, S. Nawaz, Operations on soft graphs, Fuzzy Information and Engineering 7 (2015) 423–449, doi: 10.1016/j.fiae.2015.11.003.

[7] Harary F., 1969. Graph Theory, Addison-Wesley Publishing Company.

[8] Arumugam, S. & Sundarakannan, M. (2013). Equivalence dominating sets in graphs. Utilitas Mathematica. 91.

[9] Arumugam, S., Chellali, M., & Haynes, T. W. (2013). Equivalence domination in graphs. Quaestiones Mathematicae, 36(3), 331–340. https://doi.org/10.2989/16073606.2013.779959

[10] G. Aalipour-Hafshejani, S. Akbari, Z. Ebrahimi. On D-Equivalence Class of Complete Bipartite Graphs[J], Ars Combinatoria, Volume 117. 275-288.

[11] MD, Supriya, and P. Usha. "Soft graph on distance-2 of a minimal dominating set of a graph." International Journal of Information Technology (2025): 1-6.