So I am working on this graph theory problem which says: An expansion of a graph $G$ is formed by subdividing two edges of $G$ (subdividing an edge $vw$ means replacing $vw$ with a path $vxw$ through a new vertex $x)$) and adding an edge between these 2 new vertices. Prove that if $G$ is 3-connected, then any expansion of $G$ is also 3-connected. Obtain the Peterson graph from $K_4$ by repeated expansions.
Now I was able to obtain the Peterson graph so that part is okay. For the proof I was thinking I have $G$ which is 3-connected and I obtain $G’$ by expanding $G$ (say I had edges $xy$ and $wz$ and I subdivided them to get $xsy$ and $wtz$ and the edge $st$). To prove $G’$ is 3-connected I want to show that if I remove a vertex from $G’$ I get a 2-connected graph and thus an ear decomposition. If the vertex $v$ I remove is not $s$ or $t$ then the ear decomposition for $G-v$ is supposed to give me that of $G’-v$ somehow and then I don’t know what to do with $G’-v$ when $v$ is $s$ or $t$.
Also, is there a way to do this using Menger’s theorem on internally disjoint paths?