Strengthened chain theorems for different versions of 4-connectivity

TL;DR

This paper extends chain theorems related to 4-connectivity, showing how various 4-connected graphs can be constructed from basic structures through specific operations.

Contribution

It provides strengthened chain theorems for different versions of 4-connectivity, generalizing previous results for 3-connected graphs.

Findings

Strengthened chain theorems for 4-connected graphs

Construction methods from basic graphs like $W_4$

Generalization of Tutte's chain theorem

Abstract

The chain theorem of Tutte states that every 3-connected graph can be constructed from a wheel $W_n$ by repeatedly adding edges and splitting vertices. It is not difficult to prove the following strengthening of this theorem: every non-wheel 3-connected graph can be constructed from $W_4$ by repeatedly adding edges and splitting vertices. In this paper we similarly strengthen several chain theorems for various versions of 4-connectivity.