A study of H. Martens' Theorem on chains of cycles

Let $\mathrm{\Gamma }$ be a chain of cycles of genus $g$. Let $d$,$r$ be integers with $1\le r\le g-2$ and $2r\le d\le g-3+r$. Then $w_d^r(\mathrm{\Gamma })=d-2r$ implies $\mathrm{\Gamma }$ is hyperelliptic. For each $g\ge 2r+3$ there exist non-hyperelliptic chains of cycles satisfying $w_{g-2+r}^r(\mathrm{\Gamma })=g-2-r$. In the case of algebraic curves such equality implies the curve is hyperelliptic. In particular we obtain the existence of chains of cycles $\mathrm{\Gamma }$ such that $w_{g-2+r}^r(\mathrm{\Gamma })\mathrm{\ne }{w}_{g-r}^1(\mathrm{\Gamma })$ in case $r\ge 2$. In the case of algebraic curves such numbers are equal because of the Riemann-Roch Theorem.