  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.

Combinatorics