Rational maps admit a canonical decomposition of their pullback metrics into footballs, which implies that branch data consisting of k nontrivial partitions are realizable whenever k exceeds the shortest partition length by more than one.
The Hurwitz existence problem for surface branched covers.Winter Braids Lect
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Structure and realizability for rational maps
Rational maps admit a canonical decomposition of their pullback metrics into footballs, which implies that branch data consisting of k nontrivial partitions are realizable whenever k exceeds the shortest partition length by more than one.