# Effect of collapsing independent sets on the Hadwiger number

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.

• A complete graph works vacuously. – Ilya Bogdanov Apr 18 at 22:02
• Thanks - will exclude this in the question! – Dominic van der Zypen Apr 19 at 6:15
• Such a graph would also yield a positive answer to this question of yours mathoverflow.net/questions/290435/… (same question of existence, with a weaker condition), so better stay with the other one first. BTW I would intuitively think that both are equivalent, but that might be a different question! – Wolfgang Apr 19 at 6:53