Is there a connected, finite, simple, undirected, and non-complete graph $G=(V,E)$ such that whenever an independent subset $J\subseteq V$ with $|J|>1$ is collapsed, the Hadwiger number of the resulting graph is larger than the Hadwiger number $\eta(G)$ of $G$?

**EDIT.** Excluded complete graphs; thanks to Ilya Bogdanov for his comment below.

weakercondition), so better stay with the other one first. BTW I would intuitively think that both are equivalent, but that might be a different question! $\endgroup$ – Wolfgang Apr 19 at 6:53