Kernelization and Parameterized Complexity of Star Editing and Union Editing

Abstract

The NP-hard Star Editing problem has as input a graph G = (V,E) with edges colored red and black and two positive integers k ₁ and k ₂, and determines whether one can recolor at most k ₁ black edges to red and at most k ₂ red edges to black, such that the resulting graph has an induced subgraph whose edge set is exactly the set of black edges. A generalization of Star Editing is Union Editing, which, given a hypergraph H with the vertices colored by red and black and two positive integers k ₁ and k ₂, determines whether one can recolor at most k ₁ black vertices to red and at most k ₂ red vertices to black, such that the set of red vertices becomes exactly the union of some hyperedges. Star Editing is equivalent to Union Editing when the maximum degree of H is bounded by 2. Both problems are NP-hard and have applications in chemical analytics. Damaschke and Molokov [WADS 2011] introduced another version of Star Editing, which has only one integer k in the input and asks for a solution of totally at most k recolorings, and proposed an O(k ³)-edge kernel for this new version. We improve this bound to O(k ²) and show that the O(k ²)-bound is basically tight. Moreover, we also derive a kernel with O((k ₁ + k ₂)²) edges for Star Editing. Fixed-parameter intractability results are achieved for Star Editing parameterized by any one of k ₁ and k ₂. Finally, we extend and complete the parameterized complexity picture of Union Editing parameterized by k ₁ + k ₂.