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Embedding Graphs into Embedded Graphs

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Abstract

A (possibly degenerate) drawing of a graph G in the plane is approximable by an embedding if it can be turned into an embedding by an arbitrarily small perturbation. We show that testing whether a piece-wise linear drawing of a planar graph G in the plane is approximable by an embedding can be carried out in polynomial time, if a desired embedding of G belongs to a fixed isotopy class. In other words, we show that c-planarity with embedded pipes is tractable for graphs with prescribed combinatorial embedding. To the best of our knowledge, an analogous result was previously known essentially only when G is a cycle.

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Notes

  1. In other words, a (planar) graph drawn in the plane without edge crossings.

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Acknowledgements

The author would like to thank anonymous referees for comments that helped to improve the presentation of the results, and for spotting a mistake in an earlier version of this work. The research leading to these results has received funding from the People Programme (Marie Curie Actions) of the European Union’s Seventh Framework Programme (FP7/2007-2013) under REA Grant Agreement No. [291734]. The author gratefully acknowledges support from Austrian Science Fund (FWF): M2281-N35.

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  1. University of Arizona, Tucson, AZ, USA

    Radoslav Fulek

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  1. Radoslav Fulek
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Correspondence to Radoslav Fulek.

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An extended abstract of the contribution was accepted to ISAAC 2017.

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Fulek, R. Embedding Graphs into Embedded Graphs. Algorithmica 82, 3282–3305 (2020). https://doi.org/10.1007/s00453-020-00725-3

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