Edge-Editing to a Dense and a Sparse Graph Class

We consider a graph edge-editing problem, where the goal is to transform a given graph G into a disjoint union of two graphs from a pair of given graph classes, investigating what properties of the classes make the problem fixed-parameter tractable. We focus on the case when the first class is dense, i.e. every such graph G has minimum degree at least |V (G)| − δ for a constant δ, and assume that the cost of editing to this class is fixed-parameter tractable parameterized by the cost. Under the assumptions that the second class either has bounded maximum degree, or is edge-monotone, can be defined in MSO2, and has bounded treewidth, we prove that the problem is fixed-parameter tractable parameterized by the cost. We also show that the problem is fixed-parameter tractable parameterized by degeneracy if the second class consists of independent sets and Subgraph Isomorphism is fixedparameter tractable for the input graphs. On the other hand, we prove that parameterization by degeneracy is in general W[1]-hard even for editing to cliques and independent sets.