Abstract
In the Cluster Deletion problem the goal is to remove the minimum number of edges of a given graph, such that every connected component of the resulting graph constitutes a clique. It is known that the decision version of Cluster Deletion is NP-complete on (\(P_5\)-free) chordal graphs, whereas Cluster Deletion is solved in polynomial time on split graphs. However, the existence of a polynomial-time algorithm of Cluster Deletion on interval graphs, a proper subclass of chordal graphs, remained a well-known open problem. Our main contribution is that we settle this problem in the affirmative, by providing a polynomial-time algorithm for Cluster Deletion on interval graphs. Moreover, despite the simple formulation of a polynomial-time algorithm on split graphs, we show that Cluster Deletion remains NP-complete on a natural and slight generalization of split graphs that constitutes a proper subclass of \(P_5\)-free chordal graphs. Although the later result arises from the already-known reduction for \(P_5\)-free chordal graphs, we give an alternative proof showing an interesting connection between edge-weighted and vertex-weighted variations of the problem. To complement our results, we provide faster and simpler polynomial-time algorithms for Cluster Deletion on subclasses of such a generalization of split graphs.
Highlights
In graph theoretic notions, clustering is the task of partitioning the vertices of the graph into subsets, called clusters, in such a way that there should be many edges within each cluster and relatively few edges between the clusters
It is notable that our algorithm for interval graphs, heavily relies on the linear structure obtained from their clique paths
It seems tempting to adjust our algorithm for other vertex partitioning problems on interval graphs within a more general framework, as already have been studied for particular graph properties [4, 12, 20, 21, 25]
Summary
In graph theoretic notions, clustering is the task of partitioning the vertices of the graph into subsets, called clusters, in such a way that there should be many edges within each cluster and relatively few edges between the clusters. Bonomo et al [3] characterized their optimal solution by consecutiveness of each cluster with respect to their natural ordering of the vertices Based on this fact, a dynamic programming approach led to a polynomial-time algorithm. We complement the previously-known NP-hardness of Cluster Deletion on P5-free chordal graphs, by providing a proper subclass of such graphs for which we prove that the problem remains NP-hard This result is inspired and motivated by the very simple characterization of an optimal solution on split graphs: either a maximal clique constitutes the only non-edgeless cluster, or there are exactly two non-edgeless clusters whenever there is a vertex of the independent set that is adjacent to all the vertices of the clique except one [3]. We provide a simpler and faster (linear-time) algorithm for Cluster Deletion on such graphs that avoids the usage of submodular functions minimization
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