Fig. 3

Example of a scaffold graph produced by Construction 1. Left: input graph with an independent set of size two given by the black vertices. Right: output graph with a collection of two alternating cycles and one alternating path in black. A bipartition is given by gray and white vertices. An example of a vertex-cycle is \(C_{v_1}=\{\overline{v}^1_1,v^1_1,\overline{u}_1,u_1,\overline{u}_2,u_2\}\). It is possible to turn this graph into a planar graph by replacing the edges \(\{u_3\overline{u}_2,u_2\overline{u_3},u_5\overline{u}_3\}\) with \(\{u_3u_2,u_5\overline{u_2},\overline{u}_5\overline{u}_3\}\)