Semi-Matroids on Connectivity System and Linear Decomposition

The concept of the width parameter, particularly linear-width, plays a central role in graph theory and has been extensively studied. Semimatroids, which abstract the dependence properties of affine hyperplane arrangements, have similarly garnered significant scholarly interest. This paper introduces the concept of the Semi-Matroid within a connectivity system and investigates the relationship between semimatroids on connectivity systems and linear-width. Through this exploration, we aim to contribute to a deeper understanding of structural properties and their applications in graph theoretical contexts.