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A Polynomial Kernel for Paw-Free Editing

Authors Eduard Eiben , William Lochet, Saket Saurabh

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LIPIcs.IPEC.2020.10.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Design and analysis of algorithms
  • Theory of computation → Graph algorithms analysis
  • Theory of computation → Parameterized complexity and exact algorithms
Keywords
  • Kernelization
  • Paw-free graph
  • H-free editing
  • graph modification problem

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Abstract

For a fixed graph H, the H-free Edge Editing problem asks whether we can modify a given graph G by adding or deleting at most k edges such that the resulting graph does not contain H as an induced subgraph. The problem is known to be NP-complete for all fixed H with at least 3 vertices and it admits a 2^O(k)n^O(1) algorithm. Cai and Cai [Algorithmica (2015) 71:731–757] showed that, assuming coNP ⊈ NP/poly, H-free Edge Editing does not admit a polynomial kernel whenever H or its complement is a path or a cycle with at least 4 edges or a 3-connected graph with at least one edge missing. Based on their result, very recently Marx and Sandeep [ESA 2020] conjectured that if H is a graph with at least 5 vertices, then H-free Edge Editing has a polynomial kernel if and only if H is a complete or empty graph, unless coNP ⊆ NP/poly. Furthermore they gave a list of 9 graphs, each with five vertices, such that if H-free Edge Editing for these graphs does not admit a polynomial kernel, then the conjecture is true. Therefore, resolving the kernelization of H-free Edge Editing for graphs H with 4 and 5 vertices plays a crucial role in obtaining a complete dichotomy for this problem. In this paper, we positively answer the question of compressibility for one of the last two unresolved graphs H on 4 vertices. Namely, we give the first polynomial kernel for Paw-free Edge Editing with O(k⁶) vertices.

Cite As Get BibTex

Eduard Eiben, William Lochet, and Saket Saurabh. A Polynomial Kernel for Paw-Free Editing. In 15th International Symposium on Parameterized and Exact Computation (IPEC 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 180, pp. 10:1-10:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.IPEC.2020.10

Author Details

Eduard Eiben
  • Department of Computer Science, Royal Holloway, University of London, Egham, UK
William Lochet
  • Department of Informatics, University of Bergen, Norway
Saket Saurabh
  • Institute of Mathematical Sciences, Chennai, India
  • Department of Informatics, University of Bergen, Norway

Funding

  • Lochet, William: Received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 819416).
  • Saurabh, Saket: Received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 819416), and Swarnajayanti Fellowship (No DST/SJF/MSA01/2017-18).

Acknowledgements

The authors wish to thank the anonymous referees for helping in the presentation of the paper and pointing out some missing argument in Section 5.

References

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