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Recognizing weak embeddings of graphs
Akitaya H., Fulek R., Tóth C.  ACM Transactions on Algorithms 15 (4): 1-27, 2019. Type: Article
Date Reviewed: Mar 12 2021

The paper begins:

A continuous piecewise linear map ϕ : GM is a weak embedding if, for every ε > 0, there is an embedding ψε : GM with ||ϕ-ψε|| < 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 : GH 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(mlogm) time.

After this, the authors discuss the process of “comput[ing] the combinatorial representation of an embedding ψϕ for the input ϕ.” Finally, the authors discuss how this algorithm “can be adapted to recognize weak embeddings ϕ : GH 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.

Reviewer:  Sudev Naduvath Review #: CR147212 (2106-0159)
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Graph Theory (G.2.2 )
 
 
Computational Geometry And Object Modeling (I.3.5 )
 
 
Discrete Mathematics (G.2 )
 
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