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Bounded Embeddings of Graphs in the Plane

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Combinatorial Algorithms (IWOCA 2016)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9843))

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Abstract

A drawing in the plane (\(\mathbb {R}^2\)) of a graph \(G=(V,E)\) equipped with a function \(\gamma : V \rightarrow \mathbb {N}\) is x -bounded if (i) \(x(u) <x(v)\) whenever \(\gamma (u)<\gamma (v)\) and (ii) \(\gamma (u)\le \gamma (w)\le \gamma (v)\), where \(uv\in E\) and \(\gamma (u)\le \gamma (v)\), whenever \(x(w)\in x(uv)\), where x(.) denotes the projection to the x-axis. We prove a characterization of isotopy classes of embeddings of connected graphs equipped with \(\gamma \) in the plane containing an x-bounded embedding. Then we present an efficient algorithm, which relies on our result, for testing the existence of an x-bounded embedding if the given graph is a forest. This partially answers a question raised recently by Angelini et al. and Chang et al., and proves that c-planarity testing of flat clustered graphs with three clusters is tractable when the underlying abstract graph is a forest.

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].

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Notes

  1. 1.

    By employing the technique of Bläsius and Rutter [6] also in the case of a union of internally disjoint paths between a pair of vertices [18].

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Authors and Affiliations

  1. IST Austria, Am Campus 1, 3400, Klosterneuburg, Austria

    Radoslav Fulek

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  1. Radoslav Fulek
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Correspondence to Radoslav Fulek .

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Editors and Affiliations

  1. University of Helsinki, Helsinki, Finland

    Veli Mäkinen

  2. University of Helsinki, Helsinki, Finland

    Simon J. Puglisi

  3. University of Helsinki, Helsinki, Finland

    Leena Salmela

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Fulek, R. (2016). Bounded Embeddings of Graphs in the Plane. In: Mäkinen, V., Puglisi, S., Salmela, L. (eds) Combinatorial Algorithms. IWOCA 2016. Lecture Notes in Computer Science(), vol 9843. Springer, Cham. https://doi.org/10.1007/978-3-319-44543-4_3

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  • DOI: https://doi.org/10.1007/978-3-319-44543-4_3

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