Tutte (1961): A graph $G$ is $3$-connected if and only if there exists a sequence $G_0, ...,G_n$ of graphs that have the following two properties
1) $G_0 = K_4$ and $G_n = G$
2) $G_{i+1}$ has an edge $xy$ with $d(x), d(y) ≥ 3$ and $G_i = G_{i+1}/xy$.
(If $G$ is a graph, the graph $G'$ obtained by an edge contraction of an edge $xy$ is denoted $G/xy$.)
My questions:
- How to construct the sequence?
(It seems that according to 2), we can construct by the reduction from $G_n$, however, it may can't make sure that in the final stage we can get $G_0=K_4$.)
- How to extend it to $k$-connected graph (where $k$>3)?
Reference:
[1] R. Diestel, Graph Theory, third ed., Springer-Verlag, 2005