The paper begins:

A continuous piecewise linear map ϕ : *G* → *M* is a weak embedding if, for every ε > 0, there is an embedding ψ_{ε} : *G* → *M* with ||ϕ-ψ_{ε}|| < ε.

The norm mentioned here is the uniform norm. The authors identify the challenge of recognizing a weak embedding as a combinatorial problem, and suggest a more efficient algorithm for its solution.

In the main section, the authors discuss the reconstruction and simplification of embeddings. For a graph *G* and an embedded graph *H* in an orientable surface, the main recognition algorithm starts with a piecewise linear simplicial map ϕ : *G* → *H* and then successively applies two newly introduced operations, namely clusterExpansion (Phase-1) and pipeExpansion (Phase-2), until it reports whether the simplicial map provided is a weak embedding. It is also established that the algorithm runs in *O*(*m*log*m*) time.

After this, the authors discuss the process of “comput[ing] the combinatorial representation of an embedding ψ_{ϕ} for the input ϕ.” Finally, the authors discuss how this algorithm “can be adapted to recognize weak embeddings ϕ : *G* → *H* when *H* is embedded in a nonorientable manifold.”

The authors’ approach is interesting and impressive, and the content and methodology are innovative and relevant. The graph-theoretical approach seems to be much more promising for the analysis of such combinatorial problems.