Face Covers and the Genus Problem for Apex Graphs

A graph G is an apex graph if it contains a vertex w such that G − w is a planar graph. It is easy to see that the genus g(G) of the apex graph G is bounded above by τ − 1, where τ is the minimum face cover of the neighbors of w, taken over all planar embeddings of G−w. The main result of this paper is the linear lower bound g(G) ≥ τ/160 (if G − w is 3-connected and τ > 1). It is also proved that the minimum face cover problem is NP-hard for planar triangulations and that the minimum vertex cover is NP-hard for 2-connected cubic planar graphs. Finally, it is shown that computing the genus of apex graphs is NP-hard.