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.

翻译：反馈集问题涉及从图中移除最少顶点或边以破坏其所有环。大量研究致力于理解这类问题在平面图以及有界度图上的复杂性。我们针对有界度有向图（包括平面情形）给出了这些问题的完整复杂性分类。特别地，我们证明当有向图的最大度为3时，这两个问题均为$\NP$-完全问题；而在平面有向图中，若每个顶点的入度至多为1或出度至多为1，则反馈顶点集问题可在多项式时间内求解，否则为$\NP$-完全问题。我们还给出了无向图（包括平面和非平面情形）上连通反馈顶点集问题的紧致度界。最后，我们以有界度无向图上反馈顶点集问题的历史回顾作为本文的收尾。