Generalized Graph Complementation Through Edge Partition

Graph theory is a fundamental area that explores the properties and relationships present within graph structures. This paper explores generalized graph complements by examining edge partitioning, a novel approach that extends the traditional graph complement. In analogy to the vertex partition-based definitions of generalized complements, two distinct types of generalized complements for a graph emerge through edge partitioning: k'-complement and k'(i)complement of a graph. k'-complement of a graph emerges as a versatile tool with applications across multiple domains, due to its maximum edge configuration linked with a Fibonacci polynomial and its role as a maximal outerplanar graph. The paper systematically examines theorems and fundamental properties governing the structural relationships between the original graph and its generalized complements. The characterizations of cycles and paths are also analysed in this study.