User blog comment:Wythagoras/Graphs/@comment-1605058-20140913123957/@comment-1605058-20140913161859

I don't really see how this would be executed. For example, in K3,3, we have tens of cycles, and I have no idea how we'd go to increasing length of each by two. You have to give precise procedure which would do the reduction.

Another problem would be checking what cycle is fused to what. For example, take a square with diagonal. In most obvious drawing, we have two fused cycles, but we can draw it so that we have triangle and quadrilateral sharing two edges.

Another another thing is that extending such cycles can be done in many ways - in above example, we can either add to vertices on diagonal, or one on every vertex.