The question is as follows:
Prove that applying the expansion operation* to a 3-connected graph yields a 3-connected graph.
*The expansion operation is as follows: you take two edges of a graph G, replace them by paths of length 2 through new vertices, and add an edge joining the two new vertices - the new graph is G'. (some research showed it's also called "BG-operation").
My method of solution was:
If I manage to show that G'-v for every $v\in V(G')$ is 2-connected, then G' is 3-connected. I split the cases for $v\in V(G)$, and $v \in (s,t)$ (where s,t are the two new vertices from the expansion). I then show that G'-v on both cases have ear-decomposition, based on ear-decomposition from G-v in the first case, and G-xy (where xy is the edge divided by s) in the second case.
My question is:
Are there any other methods, not using ear-decomposition to prove it?