A study of general Martens-special chains of cycles

For a general Martens-special chain of cycles $\mathrm{\Gamma }$ of type $k$ we prove that the gonality is equal to $k+2$. Although $\dim (W_{k+2}^1(\mathrm{\Gamma }))=k$ we prove that $w_{k+2}^1(\mathrm{\Gamma })=0$. We also compute the gonality sequence of $\mathrm{\Gamma }$ and we prove it is divisorial complete. We prove that a general Martens-special discrete chain of cycles $G$ of type $k$ has the same gonality sequence.