Kuratowski Chains

We prove that if H and H′ are subgraphs of a graph G, and both are isomorphic to subdivisions of K5 or K3, 3, then the following are equivalent: (i) there is a sequence H = H1, H2, ..., Hk = H′ of subgraphs of G, each isomorphic to a subdivision of K5 or K3, 3 and each differing only a "small amount" from its predecessor; (ii) H and H′ are not "separated" in G by a vertex separation of order ≤ 3. This is a lemma for use in a future paper concerning linkless embeddings of graphs in 3-space.