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:


1. 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 $G\_0=K\_4$.)


1. How to extend it to $k$-connected graph (where $k$>3)?


Reference:


[1] R. Diestel, Graph Theory, third ed., Springer-Verlag, 2005



A 3-connected graph property by Tutte
Tutte proved that every 3-connected graph with at least 5 vertices has an edge whose contraction leaves a 3-connected graph. So starting with $G\_n$, just keep contracting suitable edges until you reach $K\_4$. It is the unique 3-connected graph with less than 5 vertices, so there is no way to avoid it.


Search at Scholar for "contractible edge" to find much more on this subject, including lots of papers about generalizations to $k$-connectivity.


