Novel Idea on Edge-Ultrafilter and Edge-Tangle

The study of width parameters holds significant interest in both graph theory and algebraic settings. Among these, the tree-cut decomposition stands out as a key metric. The "Edge-tangle" concept is closely related to the "tree-cut width" width parameter in graph theory. This obstruction is often seen as vital for creating effective algorithms to calculate graph width, with the edge-tangle being the specific obstruction for tree-cut width. Meanwhile, the idea of an "Ultrafilter" is well-established in topology and algebra.  Due to their versatile nature, ultrafilters hold significant and broad-ranging importance. In this paper, we introduce a new concept called Edge-Ultrafilters for graphs and demonstrate how they are equivalent to Edge-tangles.