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Contraction Bidimensionality of Geometric Intersection Graphs

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Abstract

Given a graph G, we define \(\mathbf{bcg}(G)\) as the minimum k for which G can be contracted to the uniformly triangulated grid \(\Gamma _{k}\). A graph class \({\mathcal {G}}\) has the SQGC property if every graph \(G\in {\mathcal {G}}\) has treewidth \(\mathcal {O}(\mathbf{bcg}(G)^{c})\) for some \(1\le c<2\). The SQGC property is important for algorithm design as it defines the applicability horizon of a series of meta-algorithmic results, in the framework of bidimensionality theory, related to fast parameterized algorithms, kernelization, and approximation schemes. These results apply to a wide family of problems, namely problems that are contraction-bidimensional. Our main combinatorial result reveals a wide family of graph classes that satisfy the SQGC property. This family includes, in particular, bounded-degree string graphs. This considerably extends the applicability of bidimensionality theory for contraction bidimensional problems.

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Notes

  1. We call a set of points in the plane a body if it is homeomorphic to the closed disk \(\{(x, y) \mid x^2 + y^2 \le 1\}\). A 2-dimensional body B is a \(\alpha \)-convex if every two of its points can be the extremes of a line L consisting of \(\alpha \) straight lines and where \(L\subseteq B\). Convex bodies are exactly the 1-convex bodies.

  2. A collection of convex bodies in the pane is \(\alpha \)-fat if the ratio between the maximum and the minimum radius of a circle where all bodies of the collection can be circumscribed and inscribed respectively, is upper bounded by a.

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Funding

The first author was supported by ANR projects DEMOGRAPH (ANR-16-CE40-0028). The second and the third author were supported by the ANR projects DEMOGRAPH (ANR-16-CE40-0028), ESIGMA (ANR-17-CE23-0010), and the French-German Collaboration ANR/DFG Project UTMA (ANR-20-CE92-0027).

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Authors and Affiliations

  1. Univ Lille, Centrale Lille, CRIStAL, Lille, France

    Julien Baste

  2. LIRMM, Univ Montpellier, CNRS, Montpellier, France

    Dimitrios M. Thilikos

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  1. Julien Baste
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  2. Dimitrios M. Thilikos
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Correspondence to Dimitrios M. Thilikos.

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An extended abstract of this article appeared in the Proceedings of the 12th International Symposium on Parameterized and Exact Computation, IPEC 2017, September 6–8, 2017, Vienna, Austria [2].

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Baste, J., Thilikos, D.M. Contraction Bidimensionality of Geometric Intersection Graphs. Algorithmica 84, 510–531 (2022). https://doi.org/10.1007/s00453-021-00912-w

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