We say that a finite, simple, undirected graph $G=(V,E)$ is *contraction-sensitive* if collapsing any $2$ non-adjacent points increases the [Hadwiger number](https://en.wikipedia.org/wiki/Hadwiger_number). An example of such a graph [is the icosahedron](https://mathoverflow.net/questions/325655/contracting-non-adjacent-points-in-the-icosahedron).


Is every contraction-sensitive graph [vertex-transitive](https://en.wikipedia.org/wiki/Vertex-transitive_graph)?



Are contraction-sensitive graphs necessarily vertex-transitive?
If you pick one triangle of an icosahedron and link all its 3 vertices to an additional vertex, the resulting graph is still contraction-sensitive, as @David Roberson has kindly checked in Sage. So this is a counterexample.


