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DOI: 10.4230/LIPIcs.ICALP.2018.52
A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error

Authors: Hendrik Fichtenberger, Reut Levi, Yadu Vasudev, and Maximilian Wötzel

Published in: LIPIcs, Volume 107, 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)

Abstract
We consider one-sided error property testing of F-minor freeness in bounded-degree graphs for any finite family of graphs F that contains a minor of K_{2,k}, the k-circus graph, or the (k x 2)-grid for any k in N. This includes, for instance, testing whether a graph is outerplanar or a cactus graph. The query complexity of our algorithm in terms of the number of vertices in the graph, n, is O~(n^{2/3} / epsilon^5). Czumaj et al. (2014) showed that cycle-freeness and C_k-minor freeness can be tested with query complexity O~(sqrt{n}) by using random walks, and that testing H-minor freeness for any H that contains a cycles requires Omega(sqrt{n}) queries. In contrast to these results, we analyze the structure of the graph and show that either we can find a subgraph of sublinear size that includes the forbidden minor H, or we can find a pair of disjoint subsets of vertices whose edge-cut is large, which induces an H-minor.
Cite as

Hendrik Fichtenberger, Reut Levi, Yadu Vasudev, and Maximilian Wötzel. A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error. In 45th International Colloquium on Automata, Languages, and Programming (ICALP 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 107, pp. 52:1-52:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)

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@InProceedings{fichtenberger_et_al:LIPIcs.ICALP.2018.52,
  author =	{Fichtenberger, Hendrik and Levi, Reut and Vasudev, Yadu and W\"{o}tzel, Maximilian},
  title =	{{A Sublinear Tester for Outerplanarity (and Other Forbidden Minors) With One-Sided Error}},
  booktitle =	{45th International Colloquium on Automata, Languages, and Programming (ICALP 2018)},
  pages =	{52:1--52:14},
  series =	{Leibniz International Proceedings in Informatics (LIPIcs)},
  ISBN =	{978-3-95977-076-7},
  ISSN =	{1868-8969},
  year =	{2018},
  volume =	{107},
  editor =	{Chatzigiannakis, Ioannis and Kaklamanis, Christos and Marx, D\'{a}niel and Sannella, Donald},
  publisher =	{Schloss Dagstuhl -- Leibniz-Zentrum f{\"u}r Informatik},
  address =	{Dagstuhl, Germany},
  URL =		{https://drops-dev.dagstuhl.de/entities/document/10.4230/LIPIcs.ICALP.2018.52},
  URN =		{urn:nbn:de:0030-drops-90563},
  doi =		{10.4230/LIPIcs.ICALP.2018.52},
  annote =	{Keywords: graph property testing, minor-free graphs}
}
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