Complete soft semigraphs: comprehensive analysis of K m1−1,m2−1,...,mr−1 m1,m2,...,mr structures

Abstract

Soft set theory provides a systematic approach for handling imprecision and uncertainty by categorizing elements of a set based on specific parameters. In semigraph theory, soft semigraphs utilize this approach, offering a parameterized perspective that has significantly advanced the field through effective parameter management. In this paper, we introduce and define complete and strongly complete soft semigraphs, focusing on their unique properties and structures. We then delve into an in-depth analysis of strongly complete soft semigraphs in the form Km1−1,m2−1,...,mr−1 m1,m2,...,mr. Key properties such as the total number of f-edges, and various vertex degrees are examined through a series of theorems, providing valuable insights into the complex relationships and characteristics of these soft semigraphs.