An algorithm for delta–wye reduction of almost-planar graphs

A nonplanar graph G is almost-planar if, for every edge e of G, either G∖e or G∕e is planar. The class of almost-planar graphs were introduced in 1990 by Gubser. In 2015, Wagner proved that every almost-planar graph is delta–wye reducible to K3,3. Moreover, he showed that there exists a reduction sequence in which every graph is almost-planar. We obtain an O(n2) algorithm for Wagner’s result. We also prove that simple, 3-connected, almost-planar graphs have crossing number one and furthermore that these graphs are 3-terminal delta–wye reducible to K6 minus the edges of a triangle, whose vertices are taken as terminals.