Characterising 4-tangles through a connectivity property

TL;DR

This paper explores the relationship between connectivity properties and tangles in graphs, establishing a characterization for 4-tangles via internally 4-connected minors, thus extending known results for lower connectivity levels.

Contribution

It proves that internally 4-connected graphs have unique 4-tangles and that every graph with a 4-tangle contains an internally 4-connected minor with a corresponding tangle.

Findings

Internally 4-connected graphs have unique 4-tangles.

Graphs with a 4-tangle contain an internally 4-connected minor.

The characterization extends the understanding of tangles for k=4.

Abstract

Every large $k$-connected graph-minor induces a $k$-tangle in its ambient graph. The converse holds for $k\le 3$, but fails for $k\ge 4$. This raises the question whether `$k$-connected' can be relaxed to obtain a characterisation of $k$-tangles through highly cohesive graph-minors. We show that this can be achieved for $k=4$ by proving that internally 4-connected graphs have unique 4-tangles, and that every graph with a 4-tangle $\tau$ has an internally 4-connected minor whose unique 4-tangle lifts to $\tau$.