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

Let $C$ be a smooth curve of genus $g$ and let $d$,$r$ be integers with $1 \leq r \leq g-2$ and $2r\leq d \leq g-2+r$. H. Martens' Theorem states that $\dim (W^r_d(C))=d-2r$ implies $C$ is hyperelliptic. It is known that for a metric graph Γ of genus $g$ such statement using $\dim (W^r_d(\Gamma ))$ does not hold. However replacing $\dim (W^r_d(\Gamma ))$ by the so-called Brill-Noether rank $w^r_d(\Gamma )$ it was stated as a conjecture. Using a similar definition in the case of curves one has $\dim (W^r_d(C))=w^r_d(C)$. Let $\Gamma$ be a chain of cycles of genus $g$ and let $r,d$ be integers with $1 \leq r \leq g-2$ and $2r\leq d \leq g-3+r$. If $w^r_d(\Gamma )=d-2r$ then we prove Γ is hyperelliptic. In case $g \geq 2r+3$ then we prove there exist non-hyperelliptic chains of cycles satisfying $w^r_{g-2+r}(\Gamma )=g-2-r$, contradicting the conjecture. We give a complete description of all counterexamples within the set of chains of cycles to the statement of H. Martens' Theorem. Those counterexamples also give rise to chains of cycles such that $w^r_{g-2+r}(\Gamma ) \neq w^1_{g-r}(\Gamma )$. This shows that the Riemann-Roch duality does not hold for the Brill-Noether ranks of metric graphs.