Image contraction through fuzzy soft outerplanar graph structures

Fuzzy sets and soft sets serve as powerful mathematical tools to handle uncertainty and vagueness in real-world problems. Building on these, this study introduces the concept of fuzzy soft outerplanar graphs (FSOGs), a fusion of fuzzy soft set theory with outerplanar graph structure. In FSOGs, uncertainties are modeled by assigning graded membership values to vertices and edges through fuzzy sets, while soft set parameters allow multiple, context-dependent graph representations. This combination enables FSOGs to capture ambiguous or partially known relationships more effectively than crisp outerplanar graphs. Outerplanar graphs, a subclass of planar graphs, possess unique characteristics crucial in simplifying complex data networks. In this work, we investigate FSOGs by integrating varying degrees of membership (fuzziness) with the flexibility of soft sets, thereby generalizing crisp outerplanar graphs. We introduce VD and ED fuzzy soft outerplanar subgraphs determined by both vertex and edge deletion, including maximum and maximal fuzzy soft outerplanar subgraphs, based on vertex and edge deletions. A novel concept of the dual of fuzzy soft planar graphs is also formulated and analyzed. Theoretical results are supported through illustrative examples and formal theorems. Finally, we explored the practical application of FSOGs in image contraction.