Graph Minors. XXII. Irrelevant vertices in linkage problems

In the algorithm for the disjoint paths problem given in Graph Minors XIII, we used without proof a lemma that, in solving such a problem, a vertex which was sufficiently “insulated” from the rest of the graph by a large planar piece of the graph was irrelevant, and could be deleted without changing the problem. In this paper we prove the lemma.