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On metric graphs with prescribed gonality. (English) Zbl 1381.05012
Summary: 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 \[ \min \{3 g - 3, 2 g + 2 d - 5 \} . \] This follows from a careful parameter count to establish the upper bound and a construction of sufficiently 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 combinatorial datum capturing such harmonic maps to trees.

MSC:
05C10 Planar graphs; geometric and topological aspects of graph theory
05C12 Distance in graphs
05C05 Trees
14T05 Tropical geometry (MSC2010)
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