Editing Graphs into Disjoint Unions of Dense Clusters
In the Π-Cluster Editing problem, one is given an undirected graph G, a density measure Π, and an integer k ≥ 0, and needs to decide whether it is possible to transform G by editing (deleting and inserting) at most k edges into a dense cluster graph. Herein, a dense cluster graph is a graph in which every connected component K = (VK,EK) satisfies Π. The well-studied Cluster Editing problem is a special case of this problem with Π: =“being a clique”. In this work, we consider three other density measures that generalize cliques: 1) having at most s missing edges (s-defective cliques), 2) having average degree at least |VK| − s (average-s-plexes), and 3) having average degree at least μ· (|VK| − 1) (μ-cliques), where s and μ are a fixed integer and a fixed rational number, respectively. We first show that the Π-Cluster Editing problem is NP-complete for all three density measures. Then, we study the fixed-parameter tractability of the three clustering problems, showing that the first two problems are fixed-parameter tractable with respect to the parameter (s,k) and that the third problem is W-hard with respect to the parameter k for 0 < μ< 1.
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