  We prove that the complete graph with a hole $K_{u+w}-K_u$ can be decomposed
into cycles of arbitrary specified lengths provided that the obvious necessary
conditions are satisfied, each cycle has length at most $\min(u,w)$, and the
longest cycle is at most three times as long as the second longest. This
generalises existing results on decomposing the complete graph with a hole into
cycles of uniform length, and complements work on decomposing complete graphs,
complete multigraphs, and complete multipartite graphs into cycles of arbitrary
specified lengths.
