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The Complexity of Contracting Bipartite Graphs into Small Cycles

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Graph-Theoretic Concepts in Computer Science (WG 2022)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13453))

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Abstract

For a positive integer \(\ell \ge 3\), the \(C_\ell \)-Contractibility problem takes as input an undirected simple graph G and determines whether G can be transformed into a graph isomorphic to \(C_\ell \) (the induced cycle on \(\ell \) vertices) using only edge contractions. Brouwer and Veldman [JGT 1987] showed that \(C_4\)-Contractibility is \(\textsf{NP}\)-complete in general graphs. It is easy to verify that that \(C_3\)-Contractibility is polynomial-time solvable. Dabrowski and Paulusma [IPL 2017] showed that \(C_{\ell }\)-Contractibility is \(\textsf{NP}\)-complete on bipartite graphs for \(\ell = 6\) and posed as open problems the status of \(C_{\ell }\)-Contractibility when \(\ell \) is 4 or 5. In this paper, we show that both \(C_5\)-Contractibility and \(C_4\)-Contractibility are \(\textsf{NP}\)-complete on bipartite graphs.

The full version of this paper is at https://arxiv.org/abs/2206.07358

P. Tale—The author has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme under grant agreement SYSTEMATICGRAPH (No. 725978).

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Authors and Affiliations

  1. Indian Institute of Technology Palakkad, Palakkad, India

    R. Krithika

  2. Max Planck Institute for Informatics, Saarland Informatics Campus, Saarbrücken, Germany

    Roohani Sharma

  3. CISPA Helmholtz Center for Information Security, Saarbrücken, Germany

    Prafullkumar Tale

Authors
  1. R. Krithika
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  2. Roohani Sharma
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  3. Prafullkumar Tale
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Correspondence to Prafullkumar Tale .

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Editors and Affiliations

  1. University of Ioannina, Ioannina, Greece

    Michael A. Bekos

  2. Universität Tübingen, Tübingen, Germany

    Michael Kaufmann

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Krithika, R., Sharma, R., Tale, P. (2022). The Complexity of Contracting Bipartite Graphs into Small Cycles. In: Bekos, M.A., Kaufmann, M. (eds) Graph-Theoretic Concepts in Computer Science. WG 2022. Lecture Notes in Computer Science, vol 13453. Springer, Cham. https://doi.org/10.1007/978-3-031-15914-5_26

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  • DOI: https://doi.org/10.1007/978-3-031-15914-5_26

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