Making a Tournament Indecomposable by One Subtournament-Reversal Operation


Given a tournament T, a module of T is a subset M of V(T) such that for \(x, y\in M\) and \(v\in V(T)\setminus M\), \((v,x)\in A(T)\) if and only if \((v,y)\in A(T)\). The trivial modules of T are \(\varnothing\), \(\{u\}\) \((u\in V(T))\) and V(T). The tournament T is indecomposable if all its modules are trivial; otherwise it is decomposable. Let T be a tournament with at least five vertices. In a previous paper, the authors proved that the smallest number \(\delta (T)\) of arcs that must be reversed to make T indecomposable satisfies \(\delta (T) \le \left\lceil \frac{v(T)+1}{4} \right\rceil\), and this bound is sharp, where \(v(T) = |V(T)|\) is the order of T. In this paper, we prove that if the tournament T is not transitive of even order, then T can be made indecomposable by reversing the arcs of a subtournament of T. We denote by \(\delta '(T)\) the smallest size of such a subtournament. We also prove that \(\delta (T) = \left\lceil \frac{\delta '(T)}{2} \right\rceil\).

This is a preview of subscription content, access via your institution.


  1. 1.

    Belkhechine, H.: Decomposability index of tournaments. Discrete Math. 340, 2986–2994 (2017)

    MathSciNet  Article  Google Scholar 

  2. 2.

    Belkhechine, H., Ben Salha, C.: Decomposability and co-modular indices of tournaments. Discrete Math. 344, 112272 (2021)

    MathSciNet  Article  Google Scholar 

  3. 3.

    Belkhechine, H., Bouaziz, M., Boudabbous, I., Pouzet, M.: Inversion dans les tournois. C. R. Acad. Sci. Paris Ser. I 348, 703–707 (2010)

    MathSciNet  Article  Google Scholar 

  4. 4.

    Berge, C.: Hypergraphs—Combinatorics of Finite Sets. North-Holland, Amsterdam (1989)

    Google Scholar 

  5. 5.

    Boussaïri, A., Ille, P.: Determination of the prime bound of a graph. Contrib. Discrete Math. 9, 46–62 (2014)

    MathSciNet  MATH  Google Scholar 

  6. 6.

    Boussaïri, A., Ille, P., Woodrow, R.E.: Primitive bound of a \(2\)-structure. J. Comb. 7, 543–594 (2016)

    MathSciNet  MATH  Google Scholar 

  7. 7.

    Brignall, R.: Simplicity in relational structures and its application to permutation classes, Ph.D. Thesis, University of St Andrews, (2007)

  8. 8.

    Brignall, R., Ruškuc, N., Vatter, V.: Simple extensions of combinatorial structures. Mathematika. 57, 193–214 (2011)

    MathSciNet  Article  Google Scholar 

  9. 9.

    Ehrenfeucht, A., Rozenberg, G.: Primitivity is hereditary for 2-structures. Theoret. Comput. Sci. 70, 343–358 (1990)

    MathSciNet  Article  Google Scholar 

  10. 10.

    Erdős, P., Fried, E., Hajnal, A., Milner, E.C.: Some remarks on simple tournaments. Algebra Universalis. 2, 238–245 (1972)

    MathSciNet  Article  Google Scholar 

  11. 11.

    Erdős, P., Hajnal, A., Milner, E.C.: Simple one point extension of tournaments. Mathematika. 19, 57–62 (1972)

    MathSciNet  Article  Google Scholar 

  12. 12.

    Ille, P.: Indecomposable graphs. Discrete Math. 173, 71–78 (1997)

    MathSciNet  Article  Google Scholar 

  13. 13.

    Moon, J.W.: Embedding tournaments in simple tournaments. Discrete Math. 2, 389–395 (1972)

    MathSciNet  Article  Google Scholar 

  14. 14.

    Schmerl, J.H., Trotter, W.T.: Critically indecomposable partially ordered sets, graphs, tournaments and other binary relational structures. Discrete Math. 113, 191–205 (1993)

    MathSciNet  Article  Google Scholar 

  15. 15.

    Spinrad, J.: P4-trees and substitution decomposition. Discrete Appl. Math. 39, 263–291 (1992)

    MathSciNet  Article  Google Scholar 

Download references

Author information



Corresponding author

Correspondence to Houmem Belkhechine.

Additional information

Publisher's Note

Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.

Rights and permissions

Reprints and Permissions

About this article

Verify currency and authenticity via CrossMark

Cite this article

Belkhechine, H., Ben Salha, C. Making a Tournament Indecomposable by One Subtournament-Reversal Operation. Graphs and Combinatorics (2021).

Download citation


  • Module
  • Co-module
  • Indecomposable
  • Decomposability arc-index
  • Decomposability subtournament-index
  • Co-modular index

Mathematics Subject Classification

  • 05C20
  • 05C35