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 : V ź E ź N , a cost function c : V ź E ź N 0 and a degree function ź : V ź 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 ź ( 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 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 . We study Deletion to a Planar Graph of Given Degrees and its connected variant.These problems are known to be NP-complete and W1-hard for general graphs.We construct polynomial kernels for both problems when restricted to planar graphs.

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