Hyperelliptic graphs and metrized complexes

Abstract

We prove a version of Clifford's theorem for metrized complexes. Namely, a metrized complex that carries a divisor of degree 2r and rank r (for 0<r<g−1) also carries a divisor of degree 2 and rank 1. We provide a structure theorem for hyperelliptic metrized complexes, and use it to classify divisors of degree bounded by the genus. We discuss a tropical version of Martens' theorem for metric graphs.