Finding coherent cyclic orders in strong digraphs


A cyclic order in the vertex set of a digraph is said to be coherent if any arc is contained in a directed cycle whose winding number is one. This notion plays a key role in the proof by Bessy and Thomassé (2004) of a conjecture of Gallai (1964) on covering the vertex set by directed cycles. This paper presents an efficient algorithm for finding a coherent cyclic order in a strongly connected digraph, based on a theorem of Knuth (1974). With the aid of ear decomposition, the algorithm runs in O(nm) time, where n is the number of vertices and m is the number of arcs. This is as fast as testing if a given cyclic order is coherent.

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  1. [1]

    S. Bessy and S. Thomassé: Three min-max theorems concerning cyclic orders of strong digraphs, in Integer Programming and Combinatorial Optimization, D. Bienstock and G. Nemhauser, eds., LNCS 3064, Springer-Verlag, 2004, pp. 132–138.

  2. [2]

    D. E. Knuth: Wheels within wheels, J. Combinatorial Theory, Ser. B 16 (1974), 42–46.

    MATH  Article  MathSciNet  Google Scholar 

  3. [3]

    J. B. Orlin: A faster strongly polynomial minimum cost flow algorithm, Oper. Res. 41 (1993), 338–350.

    MATH  MathSciNet  Article  Google Scholar 

  4. [4]

    A. Schrijver: Combinatorial Optimization — Polyhedra and Efficiency, Springer-Verlag, 2003.

  5. [5]

    A. Sebő: Minmax relations for cyclically ordered digraphs, J. Combinatorial Theory, Ser. B 97(4) (2007), 518–552.

    Article  Google Scholar 

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Correspondence to Satoru Iwata.

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Iwata, S., Matsuda, T. Finding coherent cyclic orders in strong digraphs. Combinatorica 28, 83 (2008).

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Mathematics Subject Classification (2000)

  • 05C20
  • 90C27