Editing to a planar graph of given degrees

Abstract

We consider the following graph modification problem. Let the input consist of a graph G = (V,E), a weight function w: V ∪ E → N, a cost function c: V ∪ E → N and a degree function δ: V → N0, 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 δ(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 NPcomplete but have polynomial kernels when parameterized by kv + ke.