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Preprocessing Vertex-Deletion Problems: Characterizing Graph Properties by Low-Rank Adjacencies

Authors Bart M. P. Jansen , Jari J. H. de Kroon

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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Graph theory
Keywords
  • kernelization
  • vertex-deletion
  • graph modification
  • structural parameterization

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Abstract

We consider the Π-free Deletion problem parameterized by the size of a vertex cover, for a range of graph properties Π. Given an input graph G, this problem asks whether there is a subset of at most k vertices whose removal ensures the resulting graph does not contain a graph from Π as induced subgraph. Many vertex-deletion problems such as Perfect Deletion, Wheel-free Deletion, and Interval Deletion fit into this framework. We introduce the concept of characterizing a graph property Π by low-rank adjacencies, and use it as the cornerstone of a general kernelization theorem for Π-Free Deletion parameterized by the size of a vertex cover. The resulting framework captures problems such as AT-Free Deletion, Wheel-free Deletion, and Interval Deletion. Moreover, our new framework shows that the vertex-deletion problem to perfect graphs has a polynomial kernel when parameterized by vertex cover, thereby resolving an open question by Fomin et al. [JCSS 2014]. Our main technical contribution shows how linear-algebraic dependence of suitably defined vectors over 𝔽₂ implies graph-theoretic statements about the presence of forbidden induced subgraphs.

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Bart M. P. Jansen and Jari J. H. de Kroon. Preprocessing Vertex-Deletion Problems: Characterizing Graph Properties by Low-Rank Adjacencies. In 17th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 162, pp. 27:1-27:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.SWAT.2020.27

Author Details

Bart M. P. Jansen
  • Eindhoven University of Technology, The Netherlands
Jari J. H. de Kroon
  • Eindhoven University of Technology, The Netherlands

Funding

This project has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No 803421, ReduceSearch).

Acknowledgements

We would like to thank Fedor V. Fomin for hosting Jari in Bergen (Norway).

References

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