Delta–wye reduction of almost-planar graphs

A non-planar graph GG is almost planar   if, for every edge ee of GG, either G∖eG∖e or G/eG/e is planar. The main result of this paper is that every almost-planar graph is delta–wye reducible to K3,3K3,3, and moreover, there exists a reduction sequence in which every graph is almost planar. Analogous results are shown to hold for other classes of graphs, and also for regular, almost-graphic matroids.