Blocking nonorientability of a surface

Let S be a nonorientable surface. A collection of pairwise noncrossing simple closed curves in S is a blockage if every one-sided simple closed curve in S crosses at least one of them. Robertson and Thomas [9] conjectured that the orientable genus of any graph G embedded in S with sufficiently large face-width is “roughly” equal to one-half of the minimum number of intersections of a blockage with the graph. The conjecture was disproved by Mohar (Discrete Math. 182 (1998) 245) and replaced by a similar one. In this paper, it is proved that the conjectures in Mohar (1998) and Robertson and Thomas (J. Graph Theory 15 (1991) 407) hold up to a constant error term: For any graph G embedded in S, the orientable genus of G differs from the conjectured value at most by O(g2), where g is the genus of S.