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Algorithms for Outerplanar Graph Roots and Graph Roots of Pathwidth at Most 2

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Abstract

Deciding if a graph has a square root is a classical problem, which has been studied extensively both from graph-theoretic and algorithmic perspective. As the problem is NP-complete, substantial effort has been dedicated to determining the complexity of deciding if a graph has a square root belonging to some specific graph class \({{\mathcal {H}}}\). There are both polynomial-time solvable and NP-complete results in this direction, depending on \({{\mathcal {H}}}\). We present a general framework for the problem if \(\mathcal{H}\) is a class of sparse graphs. This enables us to generalize a number of known results and to give polynomial-time algorithms for the cases where \({{\mathcal {H}}}\) is the class of outerplanar graphs and \({{\mathcal {H}}}\) is the class of graphs of pathwidth at most 2.

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Notes

  1. The average degree of a graph G is defined as \(\mathrm{ad}(G)=\frac{1}{|V_G|}\sum _{v\in V_G}d_G(v)=\frac{2|E_G|}{|V_G|}\). The maximum average degree of G is then defined as \(\max \{\mathrm{ad}(H)\; |\; H\text { is a subgraph of }G\}\).

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Acknowledgements

We thank Dimitrios M. Thilikos for helpful comments on the generalizations of Theorems 1 and 2 in Sect. 6, and we thank an anonymous reviewer for helpful comments on our paper.

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Authors and Affiliations

  1. Department of Informatics, University of Bergen, PB 7803, 5020, Bergen, Norway

    Petr A. Golovach, Pinar Heggernes & Paloma T. Lima

  2. LITA, Université de Lorraine, Metz, France

    Dieter Kratsch

  3. Department of Computer Science, Durham University, Durham, DH1 3LE, UK

    Daniël Paulusma

Authors
  1. Petr A. Golovach
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  2. Pinar Heggernes
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  3. Dieter Kratsch
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  4. Paloma T. Lima
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  5. Daniël Paulusma
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Correspondence to Petr A. Golovach.

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This paper received support from the Research Council of Norway via the project “CLASSIS” (NFR grant 249994) and the Leverhulme Trust via Grant RPG-2016-258. An extended abstract of it appeared in the proceedings of WG 2017 [16].

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Golovach, P.A., Heggernes, P., Kratsch, D. et al. Algorithms for Outerplanar Graph Roots and Graph Roots of Pathwidth at Most 2. Algorithmica 81, 2795–2828 (2019). https://doi.org/10.1007/s00453-019-00555-y

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