Lemma: A disjoint neighborhood of the retained configuration

For the three-vertex graph at $(x_0,u_0,y_0)$, there is an open neighborhood $V$ whose closure misses every collision locus of that graph.

Proof. The three locations are pairwise distinct. Their pairwise Euclidean coordinate distances have a positive minimum. Choose sufficiently small vertex neighborhoods so that their closures remain pairwise disjoint. Their product is the required $V$ and misses every equality of two vertices, hence every connected collision diagonal.