Reevaluating the Complementary Relationship between Single Ultrafilters and Linear Obstacles in Connectivity Systems

The study of graph width parameters is highly significant in graph theory and combinatorics. Among these parameters, linear-width is particularly well-regarded and established. The concepts of Single Filter and Linear Obstacle pose challenges to achieving optimal linear-width in a connectivity system. In this concise paper, we present an alternative proof that establishes the cryptomorphism between Single Filter and Linear Obstacle. Although this proof may not be highly novel, we hope it will enhance the understanding of the intricate relationship between graph width parameters and ultrafilters.