Sufficient conditions for edit-optimal clusters

• We give local conditions for clusters in optimal Cluster Editing solutions.
• Bounded edit degrees is one of the sufficient conditions.
• An optimality condition for Cluster Deletion is based on degree sequences.

Cluster Editing is the problem of turning a graph into a cluster graph, that is, a disjoint union of cliques, by a minimum number of edge edits. Cluster Deletion is similarly defined, with edge deletions only. We propose a local notion of edit-optimality: Informally, we say that a crown (a certain type of labeled graph) is edit-optimal if it yields a cluster in some optimal solution to Cluster Editing, in every graph containing this crown as a subgraph. Then we give sufficient conditions for edit-optimality, e.g., in terms of vertex degrees. A condition for Cluster Deletion applies a theorem of Landau (1953) [7] on degree sequences of tournaments. The conditions are particularly suited for planted models of clusterings, and for networks that only partially exhibit a clear cluster structure.