  A new construction of BPS monodromies for 4d ${\mathcal N}=2$ theories of
class S is introduced. A novel feature of this construction is its manifest
invariance under Kontsevich-Soibelman wall crossing, in the sense that no
information on the 4d BPS spectrum is employed. The BPS monodromy is encoded by
topological data of a finite graph, embedded into the UV curve $C$ of the
theory. The graph arises from a degenerate limit of spectral networks,
constructed at maximal intersections of walls of marginal stability in the
Coulomb branch of the gauge theory. The topology of the graph, together with a
notion of framing, encode equations that determine the monodromy. We develop an
algorithmic technique for solving the equations, and compute the monodromy in
several examples. The graph manifestly encodes the symmetries of the monodromy,
providing some support for conjectural relations to specializations of the
superconformal index. For $A_1$-type theories, the graphs encoding the
monodromy are "dessins d'enfants" on $C$, the corresponding Strebel
differentials coincide with the quadratic differentials that characterize the
Seiberg-Witten curve.
