Editing to a planar graph of given degrees

Konrad K Dabrowski,Petr A Golovach,Pim Van 'T Hof,Daniël Paulusma,Dimitrios M Thilikos

doi:10.1016/j.jcss.2016.11.009

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→N0 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 NP-complete but have polynomial kernels when parameterized by kv+ke.

Highlights

Graph modification problems capture a variety of graph-theoretic problems and are well studied in algorithmic graph theory
In contrast to the aforementioned W[1]hardness results for general graphs, our two main results are that both DPGGD and DCPGGD have polynomial kernels when parameterized by kv + ke
We proved that DPGGD and DCPGGD are NP-complete but allow polynomial kernels when parameterized by kv + ke

Summary

Introduction

Graph modification problems capture a variety of graph-theoretic problems and are well studied in algorithmic graph theory. If vd ∈ S the problem is NP-complete, W[1]-hard with parameter k and FPT with parameter d + k They proved that the latter result holds even for a more general version, in which the vertices and edges have costs and the desired degree for each vertex should be in some given subset of {1, . Golovach [19] introduced a variant of Degree Constraint Editing(S) in which we insist that the resulting graph must be connected He proved that, for S = {ea}, this variant is NP-complete, FPT when parameterized by k, and has a polynomial kernel when parameterized by d + k.

Preliminaries

The Polynomial Kernels

Conclusions