Filters and Tangles in Submodular Partition Functions and Graph Width Parameters

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.