As in the paper, every pair $uv ∈ R$ has $dist_𝒫(u, v) ≤ 5$. In a $𝒫$-flip $H$, the bipartite graph $H[X, Y]$ is $G[X, Y]$ or Ḡ[X, Y], and the walk of length at most $5$ there projects to a walk in $H$ by contracting diagonal edges. A walk of length at most $r$ in $R$ then gives a walk of length at most $5r$ in every $𝒫$-flip.