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M. Kotrbcık, R. Kralovic, S. Ordyniak:
"Edge-Editing to a Dense and a Sparse Graph Class";
Talk: Latin American Theoretical Informatics Symposium, Ensenada, Mexico; 04-11-2016 - 04-15-2016; in: "Proceedings of LATIN 2016: Theoretical Informatics - 12th Latin American Symposium", (2016), ISBN: 978-3-662-49528-5; 562 - 575.

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.

http://publik.tuwien.ac.at/files/publik_257506.pdf