  Let $\Gamma$ be a chain of cycles of genus $g$. Let $d$,$r$ be integers with
$2 \leq d \leq g-1$ and $0<2r\leq d$. Then $w^r_d(\Gamma)=d-2r$ implies
$\Gamma$ is hyperelliptic unless $(d,r) = (g-1,1)$. For each $g \geq 5$ there
exist non-hyperelliptic chains of cycles satisfying $w^1_{g-1}(\Gamma)=g-3$.
The gonality $d$ of those chains of cycles ranges from $3$ to $[(g+1)/2]$. In
general they satisfy $\dim (W^1_d(\Gamma))=d-2$ but $w^1_d(\Gamma)=0$.
