Exploring two concepts: branch decomposition and weak ultrafilter on connectivity system

TL;DR

This paper investigates the duality between branch width, a graph connectivity measure, and weak ultrafilter, a logical concept, enhancing understanding in graph theory and logic.

Contribution

It introduces the concept of Weak Ultrafilter on connectivity systems and demonstrates its duality with branch decomposition.

Findings

Established duality between weak ultrafilter and branch decomposition

Enhanced understanding of connectivity and logical interpretation

Introduced new framework for connectivity systems

Abstract

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.