Filters and Tangles in Submodular Partition Functions and Graph Width Parameters
Takaaki Fujita · Asian Research Journal of Mathematics · 2025
The study of decomposition trees and graph width parameters has garnered considerable attention due to their promising applications in engineering. Within graph theory, the concept of a Loose Tangle has emerged as a notable counterpart to branch-width, a well-established graph width parameter. An ultrafilter is a maximal proper filter on a set that, for every subset, contains either the subset itself or its complement. When extended to a connectivity system, the core notions of Ultrafilter and Bramble are known to exhibit a dual relationship with branch-width. In previous studies, Filters had not been defined using submodular partition functions. This concise paper investigates the intricate relationship between Loose Tangles and Filters through the lens of a submodular partition function, a mathematical tool that embeds the principle of submodularity into the partitioning process. Furthermore, the paper explores the interaction between Brambles and Filters, also utilizing submodular partition functions as a foundational analytical framework.