On the fixed-parameter enumerability of cluster editing

Paper in proceeding, 2005

Cluster Editing is the problem of changing a graph G by at most k edge insertions or deletions into a disjoint union of cliques. The problem is motivated from computational biology and known to be fixed-parameter tractable (FPT). We study the enumeration of all solutions with a minimal set of edge changes. Enumerations can support efficient decisions between ambiguous solutions. We prove that all minimal solutions differ only on a so-called full kernel of at most k^2/4 vertices. This bound is tight. For ambiguous edges we get an optimal bound up to a constant factor. Finally we give an algorithm that outputs a compressed enumeration in O(2.4^k) time.