On the problem of finding disjoint cycles and dicycles in a digraph

Abstract

We study the following problem: Given a digraph D, decide if there is a cycle B in D and a cycle C in its underlying undirected graph UG(D) such that V (B)∩V (C)=ø.

Whereas the problem is NP-complete if, as additional part of the input, a vertex x is prescribed to be contained in C, we prove that one can decide the existence of B,C in polynomial time under the (mild) additional assumption that D is strongly connected. Our methods actually find B,C in polynomial time if they exist. The behaviour of the problem as well as our solution depend on the cycle transversal number τ (D) of D, i.e. the smallest cardinality of a set T of vertices in D such that D-T is acyclic: If τ (D)≥3 then we employ McCuaig’s framework on intercyclic digraphs to (always) find these cycles. If τ (D) = 2 then we can characterize the digraphs for which the answer is “yes” by using topological methods relying on Thomassen’s theorem on 2-linkages in acyclic digraphs. For the case τ (D)≤1 we provide an algorithm independent from any earlier work.

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References

  1. [1]

    J. Bang-Jensen and G. Gutin: Digraphs. Theory, algorithms and applications, Second edition, Springer Monographs in Mathematics, Springer-Verlag London, Ltd., London (2009).

  2. [2]

    J. Bang-Jensen and M. Kriesell: Disjoint directed and undirected paths and cycles in digraphs, Theoret. Comput. Sci. 46–49 (2009), 5138–5144.

    MathSciNet  Article  Google Scholar 

  3. [3]

    G. A. Dirac: Some results concerning the structure of graphs, Canad. Math. Bull. 6 (1963), 183–210.

    MathSciNet  MATH  Article  Google Scholar 

  4. [4]

    S. Fortune, J. Hopcroft and J. Wyllie: The directed subgraph homeomorphism problem, Theoret. Comput. Sci. 10 (1980), 111–121.

    MathSciNet  MATH  Article  Google Scholar 

  5. [5]

    L. Lovász: On graphs not containing independent circuits (in Hungarian), Mat. Lapok 16 (1965), 289–299.

    MathSciNet  MATH  Google Scholar 

  6. [6]

    W. McCuaig: Intercyclic digraphs, Graph structure theory (Seattle, WA, 1991), Contemp. Math. 147, Amer. Math. Soc., Providence, RI (1993), 203–245.

    Google Scholar 

  7. [7]

    A. Metzlar: Minimum transversal of cycles in intercyclic digraphs, PhD thesis, University of Waterloo, Ontario, Canada (1989).

  8. [8]

    A. Metzlar: Disjoint paths in acyclic digraphs, J. Combin. Theory B 57 (1993), 228–238.

    MathSciNet  MATH  Article  Google Scholar 

  9. [9]

    C. Thomassen: Disjoint cycles in digraphs, Combinatorica 3 (1983), 393–396.

    MathSciNet  MATH  Article  Google Scholar 

  10. [10]

    C. Thomassen: The 2-linkage problem for acyclic digraphs, Discrete Math. 55 (1985), 73–87.

    MathSciNet  MATH  Article  Google Scholar 

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Correspondence to Jørgen Bang-Jensen.

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Bang-Jensen, J., Kriesell, M. On the problem of finding disjoint cycles and dicycles in a digraph. Combinatorica 31, 639–668 (2011). https://doi.org/10.1007/s00493-011-2670-z

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Mathematics Subject Classification (2000)

  • 05C38
  • 05C20
  • 05C85