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Testing Outerplanarity of Bounded Degree Graphs

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Abstract

We present an efficient algorithm for testing outerplanarity of graphs in the bounded-degree model. In this model, given a graph \(G\) with degree bound \(d\), we should distinguish with high probability the case that \(G\) is outerplanar from the case that modifying at least an \(\epsilon \)-fraction of the edge set is necessary to make \(G\) outerplanar. Our algorithm runs in time polynomial in \(d\) and \(\frac{1}{\epsilon }\) only. To achieve the time complexity, we exploit the tree-like structure inherent to an outerplanar graph using the microtree/macrotree decomposition of a tree. As a corollary, we show that the property of being a cactus is testable in time polynomial in \(d\) and \(\frac{1}{\epsilon }\).

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Notes

  1. We assume that \(n\) and \(d\) are known in advance.

  2. The original definition is described in terms of the number of vertices with degree at least \(3\) instead of the number of leaves. But it is essentially equivalent to our definition.

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Acknowledgments

Hiro Ito is supported by JSPS Grants-in-Aid for Challenging Exploratory Research (No. 24650006) and MEXT Grant-in-Aid for Scientific Research on Innovative Areas (No. 24106003). Yuichi Yoshida is supported by JSPS Grant-in-Aid for Young Scientists (B) (No. 26730009), MEXT Grant-in-Aid for Scientific Research on Innovative Areas (No. 24106003), and JST, ERATO, Kawarabayashi Large Graph Project.

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Authors and Affiliations

  1. National Institute of Informatics, 2-1-2 Hitotsubashi, Chiyoda-ku, Tokyo, 101-8430, Japan

    Yuichi Yoshida

  2. Preferred Infrastructure, 2-40-1 Hongo, Bunkyo-ku, Tokyo, 113-0033, Japan

    Yuichi Yoshida

  3. The University of Electro-Communications, 1-5-1 Chofugaoka, Chofu, Tokyo, 182-8585, Japan

    Hiro Ito

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  1. Yuichi Yoshida
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  2. Hiro Ito
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Correspondence to Yuichi Yoshida.

Additional information

A preliminary version of this paper appeared in Proceedings of the 14th International Workshop on Randomization and Computation (RANDOM’10) pp. 642–655.

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Yoshida, Y., Ito, H. Testing Outerplanarity of Bounded Degree Graphs. Algorithmica 73, 1–20 (2015). https://doi.org/10.1007/s00453-014-9897-1

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  • DOI: https://doi.org/10.1007/s00453-014-9897-1

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