Partial Complementation of Graphs

Authors Fedor V. Fomin , Petr A. Golovach , Torstein J. F. Strømme , Dimitrios M. Thilikos

Thumbnail PDF


  • Filesize: 0.51 MB
  • 13 pages

Document Identifiers

Author Details

Fedor V. Fomin
  • Department of Informatics, University of Bergen, Norway
Petr A. Golovach
  • Department of Informatics, University of Bergen, Norway
Torstein J. F. Strømme
  • Department of Informatics, University of Bergen, Norway
Dimitrios M. Thilikos
  • AlGCo project-team, LIRMM, Université de Montpellier, CNRS, France. , Department of Mathematics National and Kapodistrian University of Athens, Greece

Cite AsGet BibTex

Fedor V. Fomin, Petr A. Golovach, Torstein J. F. Strømme, and Dimitrios M. Thilikos. Partial Complementation of Graphs. In 16th Scandinavian Symposium and Workshops on Algorithm Theory (SWAT 2018). Leibniz International Proceedings in Informatics (LIPIcs), Volume 101, pp. 21:1-21:13, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2018)


A partial complement of the graph G is a graph obtained from G by complementing all the edges in one of its induced subgraphs. We study the following algorithmic question: for a given graph G and graph class G, is there a partial complement of G which is in G? We show that this problem can be solved in polynomial time for various choices of the graphs class G, such as bipartite, degenerate, or cographs. We complement these results by proving that the problem is NP-complete when G is the class of r-regular graphs.

Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Theory of computation → Graph algorithms analysis
  • Partial complementation
  • graph editing
  • graph classes


  • Access Statistics
  • Total Accesses (updated on a weekly basis)
    PDF Downloads


  1. Alexandre Blanché, Konrad K. Dabrowski, Matthew Johnson, Vadim V. Lozin, Daniël Paulusma, and Viktor Zamaraev. Clique-width for graph classes closed under complementation. In Kim G. Larsen, Hans L. Bodlaender, and Jean-François Raskin, editors, 42nd International Symposium on Mathematical Foundations of Computer Science, MFCS 2017, August 21-25, 2017 - Aalborg, Denmark, volume 83 of LIPIcs, pages 73:1-73:14. Schloss Dagstuhl - Leibniz-Zentrum fuer Informatik, 2017. URL:
  2. André Bouchet. Recognizing locally equivalent graphs. Discrete Mathematics, 114(1-3):75-86, 1993. URL:
  3. Bruno Courcelle and Joost Engelfriet. Graph Structure and Monadic Second-Order Logic - A Language-Theoretic Approach, volume 138 of Encyclopedia of mathematics and its applications. Cambridge University Press, 2012. URL:
  4. Bruno Courcelle, Johann A. Makowsky, and Udi Rotics. Linear time solvable optimization problems on graphs of bounded clique-width. Theory Comput. Syst., 33(2):125-150, 2000. URL:
  5. Bruno Courcelle and Stephan Olariu. Upper bounds to the clique width of graphs. Discrete Applied Mathematics, 101(1-3):77-114, 2000. URL:
  6. Bruno Courcelle and Sang-il Oum. Vertex-minors, monadic second-order logic, and a conjecture by seese. J. Comb. Theory, Ser. B, 97(1):91-126, 2007. URL:
  7. Andrzej Ehrenfeucht, Jurriaan Hage, Tero Harju, and Grzegorz Rozenberg. Complexity issues in switching of graphs. In Hartmut Ehrig, Gregor Engels, Hans-Jörg Kreowski, and Grzegorz Rozenberg, editors, Theory and Application of Graph Transformations, 6th International Workshop, TAGT'98, Paderborn, Germany, November 16-20, 1998, Selected Papers, volume 1764 of Lecture Notes in Computer Science, pages 59-70. Springer, 1998. URL:
  8. Tomás Feder, Pavol Hell, Sulamita Klein, and Rajeev Motwani. List partitions. SIAM J. Discrete Math., 16(3):449-478, 2003. URL:
  9. M. R. Garey and David S. Johnson. Computers and Intractability: A Guide to the Theory of NP-Completeness. W. H. Freeman, 1979. Google Scholar
  10. Petr Hlinený and Sang-il Oum. Finding branch-decompositions and rank-decompositions. SIAM J. Comput., 38(3):1012-1032, 2008. URL:
  11. Vít Jelínek, Eva Jelínková, and Jan Kratochvíl. On the hardness of switching to a small number of edges. In Thang N. Dinh and My T. Thai, editors, Computing and Combinatorics - 22nd International Conference, COCOON 2016, Ho Chi Minh City, Vietnam, August 2-4, 2016, Proceedings, volume 9797 of Lecture Notes in Computer Science, pages 159-170. Springer, 2016. URL:
  12. Eva Jelínková and Jan Kratochvíl. On switching to H-free graphs. Journal of Graph Theory, 75(4):387-405, 2014. URL:
  13. Eva Jelínková, Ondrej Suchý, Petr Hlinený, and Jan Kratochvíl. Parameterized problems related to seidel’s switching. Discrete Mathematics & Theoretical Computer Science, 13(2):19-44, 2011. URL:
  14. Marcin Kaminski, Vadim V. Lozin, and Martin Milanic. Recent developments on graphs of bounded clique-width. Discrete Applied Mathematics, 157(12):2747-2761, 2009. URL:
  15. Jan Kratochvíl. Complexity of hypergraph coloring and seidel’s switching. In Hans L. Bodlaender, editor, Graph-Theoretic Concepts in Computer Science, 29th International Workshop, WG 2003, Elspeet, The Netherlands, June 19-21, 2003, Revised Papers, volume 2880 of Lecture Notes in Computer Science, pages 297-308. Springer, 2003. URL:
  16. Jan Kratochvíl, Jaroslav Nešetřil, and Ondřej Zýka. On the computational complexity of Seidel’s switching. In Fourth Czechoslovakian Symposium on Combinatorics, Graphs and Complexity (Prachatice, 1990), volume 51 of Ann. Discrete Math., pages 161-166. North-Holland, Amsterdam, 1992. URL:
  17. Sang-il Oum. Rank-width and vertex-minors. J. Comb. Theory, Ser. B, 95(1):79-100, 2005. URL:
  18. Sang-il Oum. Rank-width: Algorithmic and structural results. Discrete Applied Mathematics, 231:15-24, 2017. URL:
  19. J. J. Seidel. Graphs and two-graphs. In Proceedings of the Fifth Southeastern Conference on Combinatorics, Graph Theory and Computing (Florida Atlantic Univ., Boca Raton, Fla., 1974), volume X of Congressus Numerantium, pages 125-143. Utilitas Math., Winnipeg, Man., 1974. Google Scholar
  20. J. J. Seidel. A survey of two-graphs. In Colloquio Internazionale sulle Teorie Combinatorie (Rome, 1973), Tomo I, volume 17 of Atti dei Convegni Lincei, pages 481-511. Accad. Naz. Lincei, Rome, 1976. Google Scholar
  21. J. J. Seidel and D. E. Taylor. Two-graphs, a second survey. In Algebraic methods in graph theory, Vol. I, II (Szeged, 1978), volume 25 of Colloq. Math. Soc. János Bolyai, pages 689-711. North-Holland, Amsterdam-New York, 1981. Google Scholar
Questions / Remarks / Feedback

Feedback for Dagstuhl Publishing

Thanks for your feedback!

Feedback submitted

Could not send message

Please try again later or send an E-mail