This is a specialisation of a more general, still unanswered question.

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?

This is a specialization of a more general, still unanswered question.

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?