Editing to a Graph of Given Degrees

TL;DR

This paper introduces a fixed-parameter tractable algorithm for editing a graph to match specified degrees using vertex and edge modifications, and proves kernelization limits for the problem.

Contribution

It presents an FPT algorithm for the problem parameterized by degree bound and modification count, and shows no polynomial kernel exists unless NP is in coNP/poly.

Findings

FPT algorithm for the problem with parameter d+k

No polynomial kernel unless NP ⊆ coNP/poly

Complexity analysis of graph editing for degree constraints

Abstract

We consider the Editing to a Graph of Given Degrees problem that asks for a graph G, non-negative integers d,k and a function \delta:V(G)->{1,...,d}, whether it is possible to obtain a graph G' from G such that the degree of v is \delta(v) for any vertex v by at most k vertex or edge deletions or edge additions. We construct an FPT-algorithm for Editing to a Graph of Given Degrees parameterized by d+k. We complement this result by showing that the problem has no polynomial kernel unless NP\subseteq coNP/poly.