Graph Editing to a Given Degree Sequence

TL;DR

This paper studies the computational complexity of editing a graph to match a specified degree sequence, revealing hardness results and fixed-parameter tractability under certain conditions.

Contribution

It establishes W[1]-hardness for the problem with various operations and provides fixed-parameter algorithms and kernelization results for specific cases.

Findings

W[1]-hardness when parameterized by k for all operation combinations

FPT algorithm with time 2^{O(k(Δ+k)^2)}n^2 log n for fixed k+Δ

Polynomial kernel exists when only edge additions are allowed with parameter k+Δ

Abstract

We investigate the parameterized complexity of the graph editing problem called Editing to a Graph with a Given Degree Sequence, where the aim is to obtain a graph with a given degree sequence \sigma by at most k vertex or edge deletions and edge additions. We show that the problem is W[1]-hard when parameterized by k for any combination of the allowed editing operations. From the positive side, we show that the problem can be solved in time 2^{O(k(\Delta+k)^2)}n^2 log n for n-vertex graphs, where \Delta=max \sigma, i.e., the problem is FPT when parameterized by k+\Delta. We also show that Editing to a Graph with a Given Degree Sequence has a polynomial kernel when parameterized by k+\Delta if only edge additions are allowed, and there is no polynomial kernel unless NP\subseteq coNP/poly for all other combinations of allowed editing operations.