Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs

Authors Giacomo Paesani , Daniël Paulusma , Paweł Rzążewski

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Giacomo Paesani
  • Department of Computer Science, Durham University, UK
Daniël Paulusma
  • Department of Computer Science, Durham University, UK
Paweł Rzążewski
  • Faculty of Mathematics and Information Science, Warsaw University of Technology, Poland
  • Faculty of Mathematics, Informatics, and Mechanics, University of Warsaw, Poland


The first author thanks Carl Feghali for an inspiring initial discussion.

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Giacomo Paesani, Daniël Paulusma, and Paweł Rzążewski. Feedback Vertex Set and Even Cycle Transversal for H-Free Graphs: Finding Large Block Graphs. In 46th International Symposium on Mathematical Foundations of Computer Science (MFCS 2021). Leibniz International Proceedings in Informatics (LIPIcs), Volume 202, pp. 82:1-82:14, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2021)


We prove new complexity results for Feedback Vertex Set and Even Cycle Transversal on H-free graphs, that is, graphs that do not contain some fixed graph H as an induced subgraph. In particular, we prove that both problems are polynomial-time solvable for sP₃-free graphs for every integer s ≥ 1; here, the graph sP₃ denotes the disjoint union of s paths on three vertices. Our results show that both problems exhibit the same behaviour on H-free graphs (subject to some open cases). This is in part explained by a new general algorithm we design for finding in a graph G a largest induced subgraph whose blocks belong to some finite class C of graphs. We also compare our results with the state-of-the-art results for the Odd Cycle Transversal problem, which is known to behave differently on H-free graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Feedback vertex set
  • even cycle transversal
  • odd cactus
  • forest
  • block


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  1. Tara Abrishami, Maria Chudnovsky, Marcin Pilipczuk, Paweł Rzążewski, and Paul Seymour. Induced subgraphs of bounded treewidth and the container method. Proc. SODA 2021, pages 1948-1964, 2021. Google Scholar
  2. Yuuki Aoike, Tatsuya Gima, Tesshu Hanaka, Masashi Kiyomi, Yasuaki Kobayashi, Yusuke Kobayashi, Kazuhiro Kurita, and Yota Otachi. An improved deterministic parameterized algorithm for cactus vertex deletion. CoRR, abs/2012.04910, 2020. URL:
  3. Benjamin Bergougnoux, Édouard Bonnet, Nick Brettell, and O-Joung Kwon. Close relatives of feedback vertex set without single-exponential algorithms parameterized by treewidth. Proc. IPEC 2020, LIPIcs, 180(3):1-17, 2020. Google Scholar
  4. Édouard Bonnet, Nick Brettell, O-Joung Kwon, and Dániel Marx. Parameterized vertex deletion problems for hereditary graph classes with a block property. Proc. WG2016, LNCS, 9941:233-244, 2016. Google Scholar
  5. Andreas Brandstädt and Dieter Kratsch. On the restriction of some NP-complete graph problems to permutation graphs. Proc. FCT 1985, LNCS, 199:53-62, 1985. Google Scholar
  6. Andreas Brandstädt and Raffaele Mosca. Maximum weight independent set for l-claw-free graphs in polynomial time. Discrete Applied Mathematics, 237:57-64, 2018. Google Scholar
  7. Nina Chiarelli, Tatiana R. Hartinger, Matthew Johnson, Martin Milanič, and Daniël Paulusma. Minimum connected transversals in graphs: New hardness results and tractable cases using the price of connectivity. Theoretical Computer Science, 705:75-83, 2018. Google Scholar
  8. Marek Cygan, Fedor V. Fomin, Lukasz Kowalik, Daniel Lokshtanov, Dániel Marx, Marcin Pilipczuk, Michal Pilipczuk, and Saket Saurabh. Parameterized Algorithms. Springer, 2015. Google Scholar
  9. Konrad K. Dabrowski, Carl Feghali, Matthew Johnson, Giacomo Paesani, Daniël Paulusma, and Paweł Rzążewski. On cycle transversals and their connected variants in the absence of a small linear forest. Algorithmica, 82(10):2841-2866, 2020. Google Scholar
  10. Peter Gartland, Daniel Lokshtanov, Marcin Pilipczuk, Michał Pilipczuk, and Paweł Rzążewski. Finding large induced sparse subgraphs in C_> t -free graphs in quasipolynomial time. Proc. STOC 2021, ACM, pages 330-341, 2021. Google Scholar
  11. Sudeshna Kolay, Daniel Lokshtanov, Fahad Panolan, and Saket Saurabh. Quick but odd growth of cacti. Algorithmica, 79:271-290, 2017. Google Scholar
  12. Pranabendu Misra, Venkatesh Raman, M. S. Ramanujan, and Saket Saurabh. Parameterized algorithms for even cycle transversal. Proc. WG 2012, 7551:172-183, 2012. Google Scholar
  13. Andrea Munaro. On line graphs of subcubic triangle-free graphs. Discrete Mathematics, 340(6):1210-1226, 2017. Google Scholar
  14. Svatopluk Poljak. A note on stable sets and colorings of graphs. Commentationes Mathematicae Universitatis Carolinae, 15:307-309, 1974. Google Scholar
  15. O. Vornberger. Komplexeität von Wegeproblemen in Graphen. Reihe Theoretische Informatik, 5, 1979. Google Scholar