Let $G=(V, E)$ be an undirected simple graph.
I couldn't find the formal definition, so let's call a subset $V'\subset V$ a chunk , if it's either a singular vertex or if every pair of it's vertices are connected by a path of vertices that all belong to $V'$. Thus a chunk can be represented as a singular vertex or a tree. We say that two disjoint chunks "touch" if there is an edge connecting a vertex from one chunk to a vertex from the other chunk. If this edge is also unique, then we say that the chunks touch uniquely.
My understanding is that $G$ having the complete graph $K_n$ as minor, is equivalent to $G$ having $n$ disjoint chunks that are pairwise touching.
My question is: if $G$ has $n$ disjoint chunks that are pairwise touching, can we then also find $n$ disjoint chunks that are pairwise uniquely touching?
I have mostly looked at planar graphs and well known graphs like the Peterson to look for counterexamples, but could easily find n uniquely touching chunks in most of them. But I can't find a formal argument why this would be the case, nor a way to show it by induction. Do you think this is true, or can provide a counter example, let me know. Thanks for your time!