Tangles and Connectivity in Graphs

TL;DR

This paper introduces the concept of tangles in graphs and connectivity systems, highlighting their relationship with k-connected components and establishing a correspondence between triconnected components and tangles of order 3.

Contribution

It provides a formal link between tangles and graph connectivity, specifically proving the one-to-one correspondence for triconnected components.

Findings

Established a one-to-one correspondence between tangles of order 3 and triconnected components.

Clarified the relationship between tangles and k-connected components.

Provided foundational insights into the theory of tangles in graphs.

Abstract

This paper is a short introduction to the theory of tangles, both in graphs and general connectivity systems. An emphasis is put on the correspondence between tangles of order k and k-connected components. In particular, we prove that there is a one-to-one correspondence between the triconnected components of a graph and its tangles of order 3.