A non-planar graph G is almost planar if, for every edge e of G, either G\e or G/e is planar. The main result of this paper is that every almost-planar graph is delta-wye reducible to K3,3, 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.
Wagner, D. K. (2015). Delta-wye reduction of almost-planar graphs. Discrete Applied Mathematics, 180, 158–167. https://doi.org/10.1016/j.dam.2014.07.014