This paper explores two fundamental concepts: branch width and weak ultrafilter. Branch width is a significant graph width parameter that measures the degree of connectivity in a graph using a branch decomposition and a symmetric submodular function. Weak ultrafilter, introduced as a weakened definition of an ultrafilter, plays a vital role in interpreting defaults in logic. We introduce the concept of Weak Ultrafilter on the connectivity system (X, f) and demonstrate its duality with branch decomposition. This study enhances our understanding of these concepts in graph combinatorial and logical contexts.

翻译：本文探讨两个基本概念：分支宽度（branch width）与弱超滤（weak ultrafilter）。分支宽度是一种重要的图宽度参数，它通过分支分解和对称子模函数来度量图的连通程度。弱超滤作为超滤的弱化定义而提出，在逻辑学中对缺省解释起着关键作用。我们引入连通性系统（X, f）上弱超滤的概念，并证明其与分支分解的对偶性。本研究加深了我们对这些概念在图组合学与逻辑学语境中的理解。