Computational complexity of compaction to irreflexive cycles

In this paper, we solve a long-standing problem that has been of interest since about 1988. The problem in general is to decide whether or not it is possible to partition the vertices of a graph into k distinct non-empty sets A0, A1,..., Ak-1, such that the vertices in Ai, are independent and there is at least one edge between the pair of sets Ai, and A(i+1) mod k, for all i = 0, 1, 2,..., k - 1, k > 2, and there is no edge between any other pair of sets. Determining the computational complexity of this problem, for any value of even k ≥ 6. has been of interest since about 1988 to various people, including Pavol Hell and Jaroslav Nesetril. We show in this paper that the problem is NP-complete, for all even k ≥ 6. We study the problem as the compaction problem for an irreflexive k-cycle.