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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Author Details

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


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

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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)


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.

Subject Classification

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


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