Quasi-4-Connected Components

TL;DR

This paper introduces a new graph decomposition into quasi-4-connected components, providing a cubic time algorithm and relating it to existing tangle theory, refining previous decompositions.

Contribution

It presents a novel quasi-4-connected decomposition, along with an efficient cubic time algorithm, and establishes a connection to Robertson and Seymour's tangle theory.

Findings

Decomposition refines biconnected and triconnected components

Algorithm computes decomposition in cubic time

Establishes correspondence with tangles of order 4

Abstract

We introduce a new decomposition of a graphs into quasi-4-connected components, where we call a graph quasi-4-connected if it is 3-connected and it only has separations of order 3 that remove a single vertex. Moreover, we give a cubic time algorithm computing the decomposition of a given graph. Our decomposition into quasi-4-connected components refines the well-known decompositions of graphs into biconnected and triconnected components. We relate our decomposition to Robertson and Seymour's theory of tangles by establishing a correspondence between the quasi-4-connected components of a graph and its tangles of order 4.