  We resolve in the affirmative conjectures of Repovs and A. Skopenkov (1998),
and M. Skopenkov (2003) generalizing the classical Hanani-Tutte theorem to the
setting of approximating maps of graphs on 2-dimensional surfaces by
embeddings. Our proof of this result is constructive and almost immediately
implies an efficient algorithm for testing if a given piecewise linear map of a
graph in a surface is approximable by an embedding. More precisely, an instance
of this problem consists of (i) a graph G whose vertices are partitioned into
clusters and whose inter-cluster edges are partitioned into bundles, and (ii) a
region R of a 2-dimensional compact surface M given as the union of a set of
pairwise disjoint discs corresponding to the clusters and a set of pairwise
non-intersecting "pipes" corresponding to the bundles, connecting certain pairs
of these discs. We are to decide whether G can be embedded inside M so that the
vertices in every cluster are drawn in the corresponding disc, the edges in
every bundle pass only through its corresponding pipe, and every edge crosses
the boundary of each disc at most once.
