Suppose $$G$$ is a finite, simple graph. Let $$h(G)$$ denote the Hadwiger number, that is, the maximum $$n\in\mathbb{N}$$ such that $$K_n$$ is a minor of $$G$$.
Is there a non-complete graph $$G_0$$ with at least $$3$$ vertices and the property that whenever two non-adjacent vertices are identified, $$h(\cdot)$$ gets increased?