Feedback Set Problems on Bounded-Degree (Planar) Graphs

TL;DR

This paper classifies the computational complexity of feedback set problems on bounded-degree (planar) graphs, identifying cases where problems are NP-complete or polynomial-time solvable.

Contribution

It provides a complete complexity classification for feedback set problems on bounded-degree digraphs, including planar cases, and establishes tight degree bounds for undirected graphs.

Findings

NP-completeness for degree-3 digraphs

Polynomial solvability for certain planar digraphs

Tight degree bounds for undirected graphs

Abstract

The feedback set problems are about removing the minimum number of vertices or edges from a graph to break all its cycles. Much effort has gone into understanding their complexity on planar graphs as well as on graphs of bounded degree. We obtain a complete complexity classification for these problems on bounded-degree digraphs, including the planar case. In particular, we show that both problems are $\NP$-complete on digraphs of maximum degree three, while on planar digraphs the feedback vertex set problem is polynomial-time solvable when each vertex has either indegree at most one or outdegree at most one, and $\NP$-complete otherwise. We also give tight degree bounds for the connected feedback vertex set problem on undirected graphs, both planar and non-planar. We close the paper with a historical account of results for feedback vertex set on undirected graphs of bounded degree.