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 ε in polynomial time a kernel of order εk. This is, to the best of our knowledge, the first example of a kernelization scheme that converges to a known lower bound.
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