Editing to a Planar Graph of Given Degrees

TL;DR

This paper studies a graph modification problem on planar graphs, showing it remains NP-complete but admits polynomial kernels when parameterized by deletion weights, with applications to degree-constrained graph editing.

Contribution

It proves NP-completeness persists on planar graphs and establishes polynomial kernelization results for the problem under certain parameterizations.

Findings

NP-complete on planar graphs

Existence of polynomial kernels for the problem

Applicable to degree-constrained graph editing

Abstract

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 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 $k_v+k_e$.