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Updated: Feb 19, 2026

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Image contraction through fuzzy soft outerplanar graph structures.

Deivanai Jaisankar1, Sujatha Ramalingam2, Gizachew Bayou Zegeye3

  • 1Mathematics School of Science and Humanities, Shiv Nadar University Chennai, Rajiv Gandhi Salai (OMR), Kalavakkam, Chengalpattu, Tamil Nadu, 603110, India.

Scientific Reports
|February 17, 2026
PubMed
Summary

This study introduces fuzzy soft outerplanar graphs (FSOGs), a novel mathematical framework combining fuzzy set theory and soft sets with outerplanar graph structures to model uncertainty in complex data networks.

Keywords:
Fuzzy soft dual graphsFuzzy soft graphsFuzzy soft outerplanar graphsImage contractionMaximum and maximal fuzzy soft outerplanar subgraphsVD and ED fuzzy soft outerplanar subgraphs

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Area of Science:

  • Graph Theory
  • Set Theory
  • Computational Mathematics

Background:

  • Fuzzy sets and soft sets are established mathematical frameworks for managing uncertainty and vagueness.
  • Outerplanar graphs, a subset of planar graphs, offer simplified structures for complex data networks.
  • Existing graph models struggle to effectively represent ambiguous or partially defined relationships.

Purpose of the Study:

  • To introduce and define fuzzy soft outerplanar graphs (FSOGs) as a generalization of crisp outerplanar graphs.
  • To explore the properties and applications of FSOGs in handling uncertainty.
  • To extend the concepts of graph theory to incorporate fuzzy and soft set characteristics.

Main Methods:

  • Integration of fuzzy set theory for graded membership assignment to vertices and edges.
  • Application of soft set parameters for context-dependent graph representations.
  • Formulation of vertex deletion (VD) and edge deletion (ED) subgraphs, including maximum and maximal FSOGs.
  • Development of the dual concept for fuzzy soft planar graphs.

Main Results:

  • FSOGs effectively model ambiguous relationships by combining fuzziness and soft set flexibility.
  • New types of subgraphs (VD and ED) and dual graph concepts for fuzzy soft planar graphs are established.
  • Theoretical findings are substantiated with formal theorems and illustrative examples.
  • Demonstrated potential applications in image contraction.

Conclusions:

  • FSOGs provide a robust mathematical framework for representing uncertainty in graph structures.
  • The developed concepts offer enhanced capabilities for analyzing complex and vague data networks.
  • The study opens avenues for further research in fuzzy graph theory and its applications.