If $K_5$ or $K_{3,3}$ are homeomorphic to a subgraph of $G\circ e$, then they are homeomorphic to a subgraph of $G$.

Question

I have to proof the following:

Let $G=(V,E)$ be a graph and $e\in E$. If $G\circ e$ (contraction of $e$) contains a subgraph that is homeomorphic to $K_5$ or $K_{3,3}$, then $G$ contains a subgraph that is homeomorphic to $K_5$ or $K_{3,3}$.

My idea is to show that $G\circ e$ is a topological minor of $G$. But I don't know how to start. Any help is appreciated.

Answer 1

Hint: Here is something stronger you can prove, with fewer distractions:

Assume that $H$ is a (not necessarily topological) minor of $G$. Then --

• If $K_{3,3}$ embeds homeomorphically into $H$, then $K_{3,3}$ embeds homeomorphically into $G$, and
• If $K_5$ embeds homeomorphically into $H$ but $K_5$ does not embed homeomorphically into $G$, then $K_{3,3}$ embeds homeomorphically into $G$.