  Edge-contraction operations form an effective tool in various graph
enumeration problems, such as counting Grothendieck's dessins d'enfants and
simple and double Hurwitz numbers. These counting problems can be solved by a
mechanism known as topological recursion, which is a mirror B-model
corresponding to these counting problems. We show that for the case of orbifold
Hurwitz numbers, the mirror objects, i.e., the spectral curve and the
differential forms on it, are constructed solely from the edge-contraction
operations of the counting problem in genus $0$ and one marked point. This
forms a parallelism with Gromov-Witten theory, where genus 0 Gromov-Witten
invariants correspond to mirror B-model holomorphic geometry.

Algebraic Geometry;Combinatorics