Degenerate and Non-trivial Hyperbolic Polycycles with Two Vertices

We study some hyperbolic polycycles with two vertices which are degenerate and non-trivial and such that the maximal cyclicity in the sufficiently differentiable unfolding depends only on the vector field germ along the polycycle. In these cases, the maximal cyclicity and versal unfolding of the polycycle are described.