Cluster Editing parameterized above the size of a modification-disjoint P_3 packing is para-NP-hard
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Given a graph G=(V,E) and an integer k, the Cluster Editing problem asks whether we can transform G into a union of vertex-disjoint cliques by at most k modifications (edge deletions or insertions). In this paper, we study the following variant of Cluster Editing. We are given a graph G=(V,E), a packing H of modification-disjoint induced P_3s (no pair of P_3s in H share an edge or non-edge) and an integer ℓ. The task is to decide whether G can be transformed into a union of vertex-disjoint cliques by at most ℓ+| H| modifications (edge deletions or insertions). We show that this problem is NP-hard even when ℓ=0 (in which case the problem asks to turn G into a disjoint union of cliques by performing exactly one edge deletion or insertion per element of H). This answers negatively a question of van Bevern, Froese, and Komusiewicz (CSR 2016, ToCS 2018), repeated by Komusiewicz at Shonan meeting no. 144 in March 2019.
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