Editing to a planar graph of given degrees

We consider the following graph modification problem. Let the input consist of a graph G = (V, E), a weight function w: VUE -> N-0, a cost function c: V boolean OR E -> N-0 and a degree function 3: V -> N-0, together with three integers k(v), k(e) and C. The question is whether we can delete a set of vertices of total weight at most Icy and a set of edges of total weight at most ke so that the total cost of the deleted elements is at most C and every non deleted vertex v has degree delta(nu) in the resulting graph G'. We also consider the variant in which G' must be connected. Both problems are known to be NP-complete and W[1]-hard when parameterized by k(v), + k(e). We prove that, when restricted to planar graphs, they stay NP-complete but have polynomial kernels when parameterized by k(v) + k(e). (C) 2016 The Authors. Published by Elsevier Inc.