The Interplay between Quasi-Matroid and Connectivity Systems

A graph width parameter measures the structural complexity of a graph by assessing how it can be decomposed into simpler components. These parameters are often studied using connectivity systems that satisfy symmetric submodular conditions (Geelen et al., 2006, Oum and Seymour, 2007). Matroids are highly versatile structures with broad applications in optimization theory, combinatorial mathematics, topology, algebra, graph algorithms, game theory, geometry, and network theory, making them a subject of significant interest. Quasi-matroid, as introduced in (Kawahara, 2004) (see also (Kawahara, 2004), extend the concept of matroids by providing a more generalized framework. In this brief paper, we propose the concept of quasi-matroid on a connectivity system and explore their relationship with linear decomposition.