$3$-connected graph isomorphism problem

Question

If $G$ is $3$-connected, then $G$ contains a subgraph isomorphic to $H$, where $H$ is obtained from $K_4$ by replacing the edges of $K_4$ by internally disjoint paths.

Any hints and proofs are greatly appreciated.

Answer 1

I assume you are familiar with the fan lemma (it is not hard to prove from Menger's theorem, you should easily be able to construct a proof or find one on the internet).

First observe that $G$ must contain a cycle $C$ and a vertex $v$ that is not on $C$ (since a cycle is not 3-connected).

The fan lemma assures that there are 3 paths from $v$ to $C$ that only share the vertex $v$.

Now $v$ and the 3 endpoints of the paths on $C$ constitute the desired subdivided $K_4$, with connecting edges the cycle $C$ and the three paths from the fan lemma.