On the Hardness of Switching to a Small Number of Edges

  • Vít Jelínek
  • Eva JelínkováEmail author
  • Jan Kratochvíl
Conference paper
Part of the Lecture Notes in Computer Science book series (LNCS, volume 9797)


Seidel’s switching is a graph operation which makes a given vertex adjacent to precisely those vertices to which it was non-adjacent before, while keeping the rest of the graph unchanged. Two graphs are called switching-equivalent if one can be made isomorphic to the other one by a sequence of switches.

Jelínková et al. [DMTCS 13, no. 2, 2011] presented a proof that it is NP-complete to decide if the input graph can be switched to contain at most a given number of edges. There turns out to be a flaw in their proof. We present a correct proof.

Furthermore, we prove that the problem remains NP-complete even when restricted to graphs whose density is bounded from above by an arbitrary fixed constant. This partially answers a question of Matoušek and Wagner [Discrete Comput. Geom. 52, no. 1, 2014].


Seidel’s switching Computational complexity Graph density Switching-minimal graphs NP-completeness 


  1. 1.
    Bui, T.N., Chaudhuri, S., Leighton, F.T., Sipser, M.: Graph bisection algorithms with good average case behavior. Combinatorica 7(2), 171–191 (1987)MathSciNetCrossRefGoogle Scholar
  2. 2.
    Ehrenfeucht, A., Hage, J., Harju, T., Rozenberg, G.: Complexity issues in switching of graphs. In: Ehrig, H., Engels, G., Kreowski, H.-J., Rozenberg, G. (eds.) TAGT 1998. LNCS, vol. 1764, pp. 59–70. Springer, Heidelberg (2000)CrossRefGoogle Scholar
  3. 3.
    Garey, M.R., Johnson, D.S., Stockmeyer, L.: Some simplified NP-complete graph problems. Theor. Comput. Sci. 1(3), 237–267 (1976)MathSciNetCrossRefzbMATHGoogle Scholar
  4. 4.
    Hage, J.: Structural Aspects of Switching Classes. Ph.D. thesis, Leiden Institute of Advanced Computer Science (2001)Google Scholar
  5. 5.
    Jelínková, E., Kratochvíl, J.: On switching to H-free graphs. J. Graph Theor. 75(4), 387–405 (2014)MathSciNetCrossRefzbMATHGoogle Scholar
  6. 6.
    Jelínková, E., Suchý, O., Hliněný, P., Kratochvíl, J.: Parameterized problems related to Seidel’s switching. Discrete Math. Theor. Comput. Sci. 13(2), 19–42 (2011)MathSciNetzbMATHGoogle Scholar
  7. 7.
    Kozerenko, S.: On graphs with maximum size in their switching classes. Comment. Math. Univ. Carol. 56(1), 51–61 (2015)MathSciNetzbMATHGoogle Scholar
  8. 8.
    Kratochvíl, J.: Complexity of hypergraph coloring and Seidel’s switching. In: Bodlaender, H.L. (ed.) WG 2003. LNCS, vol. 2880, pp. 297–308. Springer, Heidelberg (2003)CrossRefGoogle Scholar
  9. 9.
    Lindzey, N.: Speeding up graph algorithms via switching classes. In: Kratochvíl, J., Miller, M., Froncek, D. (eds.) IWOCA 2014. LNCS, vol. 8986, pp. 238–249. Springer, Heidelberg (2015)CrossRefGoogle Scholar
  10. 10.
    Matoušek, J., Wagner, U.: On Gromov’s method of selecting heavily covered points. Discrete Comput. Geom. 52(1), 1–33 (2014)MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

© Springer International Publishing Switzerland 2016

Authors and Affiliations

  • Vít Jelínek
    • 1
  • Eva Jelínková
    • 2
    Email author
  • Jan Kratochvíl
    • 2
  1. 1.Computer Science Institute, Faculty of Mathematics and PhysicsCharles UniversityPrahaCzech Republic
  2. 2.Department of Applied Mathematics, Faculty of Mathematics and PhysicsCharles UniversityPrahaCzech Republic

Personalised recommendations