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 $w:V\cup E\rightarrow \mathbb {N}$, a cost function $c:V\cup E\rightarrow \mathbb {N}$ and a degree function $\delta :V\rightarrow \mathbb {N}_0$, 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 $\mathsf{NP}$-complete and $\mathsf{W}[1]$-hard when parameterized by $k_v+k_e$. We prove that, when restricted to planar graphs, they stay $\mathsf{NP}$-complete but have polynomial kernels when parameterized by $k_v+k_e$.

The first and fourth author were supported by EPSRC Grant EP/K025090/1. The research of the second author has received funding from the European Research Council under the European Union’s Seventh Framework Programme (FP/2007–2013)/ERC Grant Agreement n. 267959. The research of the fifth author was co-financed by the European Union (European Social Fund ESF) and Greek national funds through the Operational Program “Education and Lifelong Learning” of the National Strategic Reference Framework (NSRF) - Research Funding Program: ARISTEIA II.