Computational Complexity of Compaction to Reflexive Cycles

Authors:
Narayan Vikas
Affiliations:
-
Venue:
SIAM Journal on Computing
Year:
2003

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Abstract

In this paper, we solve a widely publicized open problem posed by Peter Winkler in 1988. The problem is to decide whether or not it is possible to partition the vertices of a graph into four distinct nonempty sets A, B, C, and D, such that there is no edge between the sets A and C, and between the sets B and D, and that there is at least one edge between any other pair of distinct sets. Winkler asked whether this problem is NP-complete. We show in this paper that it is NP-complete. We study the problem as the compaction problem for a reflexive 4-cycle. We also show in this paper that the compaction problem for a reflexive k-cycle is NP-complete for all $k \geq 4$.