EXPLORING PLANARITY AND MINOR THEORY IN FRAMED 4-VALENT GRAPHS: A PATH TOWARD STRUCTURAL CLASSIFICATION

Abstract

The structural properties of framed 4-valent graphs are studied in this work with specific focus on minor theory and planarity, which are fundamental to graph transformation and classification. Planarity restrictions are determined in the study by the identification of forbidden minors, e.g., K₅ and K₃,₃ via Kuratowski's Theorem. To identify minor-minimal structures that are minimally non-planar, minor theory is applied to study graph transformations via edge contractions and deletions. The research provides a computer science structure for analyzing network connectivity, symmetry, and categorization through the introduction of matrix representations in the form of adjacency matrices. For easing graphs while preserving important structural properties, quotient graphs and equivalence relations are also explored. It also helps advance topological graph theory, network optimization, and algorithmic categorization methods by enhancing our knowledge on framed 4-valent graphs through the combination of theoretical models and representative frameworks. Future research into computer-aided planarity testing and big data graph analysis will be enabled by the findings, which have significant implications for computational topology, algebraic graph theory, and efficient network design.