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 $w\colon V\cup E\rightarrow \mathbb{N}$, a cost function $c\colon V\cup E\rightarrow \mathbb{N}$ and a degree function $\delta\colon V\rightarrow \mathbb{N}0$, together with three integers $kv, ke$ and $C$. The question is whether we can delete a set of vertices of total weight at most $kv$ 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 $\delta(v)$ 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 $kv+ke$. We prove that, when restricted to planar graphs, they stay NP-complete but have polynomial kernels when parameterized by $kv+k_e$.