Abstract. Directed Feedback Vertex Set (DFVS) is a fundamental computational problem that has received a lot of attention in parameterized complexity. In this paper, we initiate the study of a wide generalization of this problem called the [Formula: see text]-free Strong Connected Component Deletion problem, where [Formula: see text] is a finite family of digraphs. Here, one is given a digraph [Formula: see text] and an integer [Formula: see text], and the objective is to decide whether there is a vertex set of size at most [Formula: see text] whose deletion results in a digraph where every strongly connected component excludes graphs in family [Formula: see text] as (not necessarily induced) subgraphs. When [Formula: see text] comprises only the digraph with a single arc, then this problem is precisely the DFVS problem. Our main result is a proof that this problem is fixed-parameter tractable parameterized by the size of the deletion set if [Formula: see text] only contains rooted graphs or if [Formula: see text] contains at least one directed path. Along with generalizing the fixed-parameter tractability result for DFVS, our result also generalizes the results of Göke, Marx, and Mnich [ Proceedings of the International Conference on Algorithms and Complexity, Springer, 2019, pp. 249–261] for the 1-Out-Regular Vertex Deletion and Bounded Size Strong Component Vertex Deletion problems. Moreover, we design algorithms for the two above-mentioned problems, whose running times are better and that match with the best bounds for DFVS, without using the heavy machinery of shadow removal as is done by Göke, Marx, and Mnich [ Proceedings of the International Conference on Algorithms and Complexity, Springer, 2019, pp. 249–261].

Directed Feedback Vertex Set (DFVS) is a fundamental computational problem that has received extensive attention in parameterized complexity. In this paper, we initiate the study of a wide generalization, the ℋ-SCC Deletion problem. Here, one is given a digraph D, an integer k and the objective is to decide whether there is a vertex set of size at most k whose deletion leaves a digraph where every strong component excludes graphs in the fixed finite family ℋ as (not necessarily induced) subgraphs. When ℋ comprises only the digraph with a single arc, then this problem is precisely DFVS. Our main result is a proof that this problem is fixed-parameter tractable parameterized by the size of the deletion set if ℋ only contains rooted graphs or if ℋ contains at least one directed path. Along with generalizing the fixed-parameter tractability result for DFVS, our result also generalizes the recent results of Göke et al. [CIAC 2019] for the 1-Out-Regular Vertex Deletion and Bounded Size Strong Component Vertex Deletion problems. Moreover, we design algorithms for the two above mentioned problems, whose running times are better and match with the best bounds for DFVS, without using the heavy machinery of shadow removal as is done by Göke et al. [CIAC 2019].