Suppose $G$ is a finite simple graph. Let $h(G)$ denote the *Hadwiger number* (synonym: *connected pseudoachromatic number*, cf. [Abrams-Berman 2014, p. 315]) of $G$; 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 with the following two properties?

- whenever you remove a vertex, the chromatic number gets decreased (that is, $G$ is
vertex-critical, to use an established term), and- whenever you identify two non-adjacent vertices, $h(\cdot)$ gets increased. (We could call graphs with this property "
identification-critical"; I do not know of an established terminology).

**References**

[Abrams-Berman 2014] Lowell Abrams, Yosef Berman, ''Connected pseudoachromatic index of complete graphs'', Australasian Journal of Combinatorics, Volume 60(3) (2014), 314-324