Feedback Set Problems on Bounded-Degree (Planar) Graphs

2026-05-11

AI Summary

This study investigates the computational complexity of the Feedback Vertex Set and Feedback Edge Set problems in bounded-degree and planar graphs. Through graph-theoretic reductions, structural decompositions, and fine-grained complexity analysis, it provides the first complete characterization of the complexity landscape for both problems on directed graphs, including the planar case. The main contributions include proving that both problems remain NP-complete even when restricted to directed graphs of maximum degree three. For planar directed graphs, the Feedback Vertex Set problem is shown to be polynomial-time solvable if every vertex has either in-degree or out-degree at most one, but becomes NP-complete otherwise. Additionally, the work establishes tight degree bounds for the connected Feedback Vertex Set problem in 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.

Problem

• Feedback Set
• Bounded-Degree Graphs
• Planar Graphs
• NP-completeness
• Computational Complexity

Innovation

• feedback vertex set
• bounded-degree graphs
• planar digraphs
• computational complexity
• NP-completeness