On metric graphs with prescribed gonality

Cools, Filip; Draisma, Jan (2018). On metric graphs with prescribed gonality. Journal of combinatorial theory. Series A, 156, pp. 1-21. Elsevier 10.1016/j.jcta.2017.11.017

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We prove that in the moduli space of genus-g metric graphs the
locus of graphs with gonality at most d has the classical dimension
min{3g - 3; 2g + 2d - 5g}:
This follows from a careful parameter count to establish the upper bound and
a construction of suffciently many graphs with gonality at most d to establish
the lower bound. Here, gonality is the minimal degree of a non-degenerate harmonic
map to a tree that satisfies the Riemann-Hurwitz condition everywhere.
Along the way, we establish a convenient

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