Kernelization and Parameterized Complexity of Star Editing and Union Editing

The NP-hard Star Editing problem has as input a graph G = (V, E) with edges colored red and black and two positive integers k1 and k2, and determines whether one can recolor at most k1 black edges to red and at most k2 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 k1 and k2, determines whether one can recolor at most k1 black vertices to red and at most k2 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)-edge kernel for this new version. We improve this bound to O(k) and show that the O(k)-bound is basically tight. Moreover, we also derive a kernel with O((k1 + k2)) edges for Star Editing. Fixed-parameter intractability results are achieved for Star Editing parameterized by any one of k1 and k2. Finally, we extend and complete the parameterized complexity picture of Union Editing parameterized by k1 + k2.