Editing Graphs Into Few Cliques: Complexity, Approximation, and Kernelization Schemes

Given an undirected graph G and a positive integer k, the NP-hard Sparse Split Graph Editing problem asks to transform G into a graph that consists of a clique plus isolated vertices by performing at most k edge insertions and deletions; similarly, the \(P_3\)-Bag Editing problem asks to transform G into a graph which is the union of two possibly overlapping cliques. We give a simple linear-time 3-approximation algorithm for Sparse Split Graph Editing, an improvement over a more involved known factor-3.525 approximation. Further, we show that \(P_3\)-Bag Editing is NP-complete. Finally, we present a kernelization scheme for both problems and additionally for the 2-Cluster Editing problem. This scheme produces for each fixed \(\varepsilon \) in polynomial time a kernel of order \(\varepsilon k\). This is, to the best of our knowledge, the first example of a kernelization scheme that converges to a known lower bound.