Let $G=(V,E)$ be a connected, simple, finite, undirected graph. The Hadwiger number $\eta(G)$ is the maximum integer $n\in \mathbb{N}$ such that $K_n$ is a minor of $G$.
We say that $G$ is contraction-Hadwiger-critical (cHc) if contracting any pair of non-adjacent vertices increases the Hadwiger number. (The icosahedron graph is an example of a contraction-critical graph.)
Morever, we call $G$ edge-Hadwiger-critical (eHc) if adding an edge between any pair of non-adjacent vertices increases the Hadwiger number.
It is not hard to show that any cHc graph is eHc.
Does the converse hold?
