Optimality of the bounded-genus face-cover bound

Determine whether the O(g t^4) upper bound on the size of a face cover for a 3-connected rooted graph of Euler genus g and sufficiently large face-width, without a rooted K_{2,t} minor, is optimal.

Background

For rooted graphs embedded in a surface of Euler genus g with sufficiently large face-width, the paper derives a face-cover bound of O(g·t^4) when no rooted K_{2,t} minor is present. The dependence on g and t follows from reducing the bounded-genus setting to planar and projective-planar pieces. The authors do not establish whether this resulting bound has the optimal asymptotic dependence.