Apex graphs with embeddings of face-width three

An apex graph is a graph which has a vertex whose removal makes the resulting graph planar. Embeddings of apex graphs having face-width three are characterized. Surprisingly, there are such embeddings of arbitrarily large genus. This solves a problem of Robertson and Vitray. We also give an elementary proof of a result of Robertson, Seymour, and Thomas (1993) that any embedding of an apex graph in a nonorientable surface has face-width at most two.