Nonorientability of a Surface

Let S be a nonorientable surface. A collection of pairwise noncross-ing simple closed curves in S is a blockage if every onesided 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 suuciently 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 7] and replaced by a similar one. In this paper, it is proved that the conjectures in 7, 9] hold up to a constant error term: For any graph G embedded in S, the orientable genus of G diiers from the conjectured value at most by O(g 2), where g is the genus of S.