Various Properties of Various Ultrafilters, Various Graph Width Parameters, and Various Connectivity Systems (with Survey)
Takaaki Fujita
TL;DR
This work develops a unifying framework for ultrafilters on connectivity systems, defined as pairs $(X,f)$ with a finite ground set $X$ and a symmetric submodular function $f$, and investigates their dual relationship with graph width parameters such as branch-width and linear branch-width. It introduces foundational concepts—filters, ultrafilters, prefilters, subbases, tangles, brambles, and chain/antichain structures—within the connectivity-system setting, and establishes key dualities (e.g., between branch-width and ultrafilters) alongside Tukey's lemma applicability for existence results. The paper also surveys numerous width/length/depth parameters, proposes new linear and directed variants, and outlines extensive future tasks linking ultrafilters to advanced decompositions, matroid theory, fuzzy structures, and directed hypergraphs. By tying set-theoretic tools (ultrafilters, Tukey's lemma) to combinatorial width notions across finite and infinite contexts, it lays groundwork for cross-disciplinary techniques in graph theory, topology, and logic with potential computational implications. The work is largely conceptual and methodological, setting a broad agenda for future rigorous developments and algorithmic investigations in graph width parameters and their ultrafilter obstructions.
Abstract
This paper investigates ultrafilters in the context of connectivity systems, defined as pairs $(X, f)$ where $X$ is a finite set and $f$ is a symmetric submodular function. Ultrafilters, essential in topology and set theory, are extended to these systems, with a focus on their relationship to graph width parameters, which help analyze graph complexity. We demonstrate theorems for ultrafilters on connectivity systems and explore related concepts such as prefilters, ultra-prefilters, and subbases. New parameters for width, length, and depth are introduced, providing further insight into graph width. The study also includes a comparison of various graph width parameters and their related concepts, offering a foundation for future research in graph theory and computational complexity. Additionally, we explore connections to other mathematical disciplines, including set theory, lattice theory, and matroid theory, expanding the scope of ultrafilters and graph width. (It also includes information similar to that found in surveys, aiming to promote future research on graph width parameters.)