Cycle-minors and subdivisions of wheels

In this paper we prove two results. The first is an extension of a result of Dirac which says that any set of n vertices of an n-connected graph lies in a cycle. We prove that if V ′ is a set of at most 2n vertices in an n-connected graph G, then G has, as a minor, a cycle using all of the vertices of V′. The second result says that if G is an n+1-connected graph with maximum vertex degree then G contains a subgraph that is a subdivision of W2n if and only if ≥2n. 2009 Wiley Periodicals, Inc. J Graph Theory 62: 100–108, 2009