Blocking nonorientability of a surface
A collection of pairwise noncrossing simple closed curves in a nonorientable surface \(S\) is a blockage if every one-sided simple closed curve in \(S\) crosses at least one of them. \textit{N. Robertson} and \textit{R. Thomas} [J. Graph Theory 15, 407-419 (1991; Zbl 0735.05031)] conjectured that the orientable genus of any graph \(G\) imbedded 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. \textit{B. Mohar} [Discrete Math. 182, 245-253 (1998; Zbl 0889.05040)] disproved that conjecture, and made is revised one. In the present paper, the authors show that the two conjectures hold up to a constant error term: For any graph \(G\) imbedded in \(S\), the orientable genus of \(G\) differs from the conjectured value by at most \(O(g^2)\), where \(S\) has genus \(g\).
scientific article; zbMATH DE number 1967935