Minimum cycle factors in quasi-transitive digraphs

We consider the minimum cycle factor problem: given a digraph D, find the minimum number kmin(D) of vertex disjoint cycles covering all vertices of D or verify that D has no cycle factor. There is an analogous problem for paths, known as the minimum path factor problem. Both problems are NP-hard for general digraphs as they include the Hamilton cycle and path problems, respectively. In 1994 Gutin [G. Gutin, Polynomial algorithms for finding paths and cycles in quasi-transitive digraphs, Australas. J. Combin. 10 (1994) 231–236] proved that the minimum path factor problem is solvable in polynomial time, for the class of quasi-transitive digraphs, and so is the Hamilton cycle problem. As the minimum cycle factor problem is analogous to the minimum path factor problem and is a generalization of the Hamilton cycle problem, it is therefore a natural question whether this problem is also polynomially solvable, for quasi-transitive digraphs. We conjecture that the problem of deciding, for a fixed k, whether a quasi-transitive digraph D has a cycle factor with at most k cycles is polynomial, and we verify this conjecture for k = 3. We introduce the notion of an irreducible cycle factor and show how to convert a given cycle factor into an irreducible one in polynomial time when the input digraph is quasi-transitive. Finally, we show that even though this process will often reduce the number of cycles considerably, it does not always yield a minimum cycle factor. c © 2007 Elsevier B.V. All rights reserved.