Finding coherent cyclic orders in strong digraphs

Abstract

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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Correspondence to Satoru Iwata.

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Iwata, S., Matsuda, T. Finding coherent cyclic orders in strong digraphs. Combinatorica 28, 83 (2008). https://doi.org/10.1007/s00493-008-2305-1

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Mathematics Subject Classification (2000)

  • 05C20
  • 90C27