Résumé
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 $k_v$, $k_e$ and $C$. The question is whether we can delete a set of vertices of total weight at most $k_v$ and a set of edges of total weight at most $k_e$ 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 \classNP-complete and \classW{1}-hard when parameterized by $k_v+k_e$. We prove that, when restricted to planar graphs, they stay \classNP-complete but have polynomial kernels when parameterized by $k_v+k_e$.