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Reducing Graph Transversals via Edge Contractions

Authors Paloma T. Lima, Vinicius F. dos Santos , Ignasi Sau , Uéverton S. Souza

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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
  • blocker problem
  • edge contraction
  • graph transversal
  • parameterized complexity
  • vertex cover
  • feedback vertex set
  • odd cycle transversal

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Abstract

For a graph parameter π, the Contraction(π) problem consists in, given a graph G and two positive integers k,d, deciding whether one can contract at most k edges of G to obtain a graph in which π has dropped by at least d. Galby et al. [ISAAC 2019, MFCS 2019] recently studied the case where π is the size of a minimum dominating set. We focus on graph parameters defined as the minimum size of a vertex set that hits all the occurrences of graphs in a collection ℋ according to a fixed containment relation. We prove co-NP-hardness results under some assumptions on the graphs in ℋ, which in particular imply that Contraction(π) is co-NP-hard even for fixed k = d = 1 when π is the size of a minimum feedback vertex set or an odd cycle transversal. In sharp contrast, we show that when π is the size of a minimum vertex cover, the problem is in XP parameterized by d.

Cite As Get BibTex

Paloma T. Lima, Vinicius F. dos Santos, Ignasi Sau, and Uéverton S. Souza. Reducing Graph Transversals via Edge Contractions. In 45th International Symposium on Mathematical Foundations of Computer Science (MFCS 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 170, pp. 64:1-64:15, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.MFCS.2020.64

Author Details

Paloma T. Lima
  • Department of Informatics, University of Bergen, Norway
Vinicius F. dos Santos
  • Departamento de Ciência da Computação, Universidade Federal de Minas Gerais, Belo Horizonte, Brazil
Ignasi Sau
  • LIRMM, Université de Montpellier, CNRS, France
Uéverton S. Souza
  • Instituto de Computação, Universidade Federal Fluminense, Niterói, Brazil

Funding

  • dos Santos, Vinicius F.: Grant APQ-02592-16 Minas Gerais Research Support Foundation (FAPEMIG) and Grants 311679/2018-8 and 421660/2016-3 National Council for Scientific and Technological Development (CNPq).
  • Sau, Ignasi: CAPES-PRINT Institutional Internationalization Program, process 88887.371209/ 2019-00, and projects DEMOGRAPH (ANR-16-CE40-0028) and ESIGMA (ANR-17-CE23-0010).
  • Souza, Uéverton S.: Grant E-26/203.272/2017 Rio de Janeiro Research Support Foundation (FAPERJ) and Grant 303726/2017-2 National Council for Scientific and Technological Development (CNPq).

Acknowledgements

We would like to thank the anonymous reviewers for helpful remarks that improved the presentation of the manuscript.

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