Face covers and the genus problem for apex graphs

Abstract

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. © 2001 Academic Press.