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

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