This book studies ultrafilters on connectivity systems, that is, on pairs \((X,f)\) where \(X\) is a finite set and \(f:2^{X}\to \mathbb{N}\) is a symmetric submodular function. Ultrafilters, which play a fundamental role in topology and set theory, are considered here in this broader setting, with particular emphasis on their connections to graph width parameters and to the structural analysis of graph complexity. We develop several results on ultrafilters on connectivity systems and examine related notions such as prefilters, ultra-prefilters, and filter subbases. We also discuss additional width-, length-, and depth-type parameters that naturally arise in this framework, thereby broadening the perspective from which graph structure may be studied. In addition, the book compares a wide range of graph width parameters and related concepts, with the aim of providing a unified viewpoint and a useful point of departure for further research in graph theory and computational complexity. More broadly, the book highlights connections with several neighboring areas of mathematics, including set theory, lattice theory, and matroid theory. It also contains survey-style material intended to clarify the current landscape of graph width theory and to stimulate further developments in the subject.

翻译：本书研究连通性系统上的超滤子，即研究形如 \((X,f)\) 的对，其中 \(X\) 为有限集，\(f:2^{X}\to \mathbb{N}\) 为对称子模函数。在拓扑学与集合论中扮演基础角色的超滤子，在此更广泛的设定下被考察，尤其关注其与图宽参数及图复杂度结构分析的联系。我们建立了若干关于连通性系统上超滤子的结果，并探讨了相关概念，如前滤子、超前滤子与滤子子基。同时，讨论了在该框架下自然出现的其他宽度型、长度型与深度型参数，从而拓宽了研究图结构的视角。此外，本书比较了广泛的图宽参数及相关概念，旨在提供统一视角，并为图论与计算复杂性的进一步研究提供有益的出发点。更广泛而言，本书强调了与多个邻近数学领域（包括集合论、格论与拟阵论）的联系。书中还包含综述性材料，旨在阐明图宽理论当前的研究格局，并推动该领域的进一步发展。