  A branched covering map of surfaces induces a map in the opposite direction
between their arc complexes. We represent a branched covering map
combinatorially using what we call a lifting picture, and use this
representation to computably solve the membership problem of the set of
weighted arc diagrams on a given surface which can be obtained by lifting a
weighted arc diagram on a bigon. We provide a brute force solution in the
general case, and an efficient solution when the input arc diagram is a
triangulation.
