The Directed Feedback Vertex Set (DFVS) problem takes as input a directed graph G and seeks a smallest vertex set S that hits all cycles in G. This is one of Karp's 21 NP-complete problems. Resolving the parameterized complexity status of DFVS was a long-standing open problem until Chen et al. [STOC 2008, J. ACM 2008 showed its fixedparameter tractability via a 4 k k!n O(1) -time algorithm, where k = |S|.Here we show fixed-parameter tractability of two generalizations of DFVS:• Find a smallest vertex set S such that every strong component of G − S has size at most s: we give an algorithm solving this problem in time 4 k (ks + k + s)! · n O(1) . This generalizes an algorithm by Xiao [JCSS 2017] for the undirected version of the problem.• Find a smallest vertex set S such that every non-trivial strong component of G − S is 1-out-regular: we give an algorithm solving this problem in time 2 O(k 3 ) · n O(1) .We also solve the corresponding arc versions of these problems by fixed-parameter algorithms.

A classical branch of graph algorithms is graph transversals, where one seeks a minimum-weight subset of nodes in a node-weighted graph G which intersects all copies of subgraphs F from a fixed family F. Many such graph transversal problems have been shown to admit polynomial-time approximation schemes (PTAS) for planar input graphs G, using a variety of techniques like the shifting technique (Baker, J. ACM 1994), bidimensionality (Fomin et al., SODA 2011), or connectivity domination (Cohen-Addad et al., STOC 2016). These techniques do not seem to apply to graph transversals with parity constraints, which have recently received significant attention, but for which no PTASs are known.In the even-cycle transversal (ECT) problem, the goal is to find a minimum-weight hitting set for the set of even cycles in an undirected graph. For ECT, Fiorini et al. (IPCO 2010) showed that the integrality gap of the standard covering LP relaxation is Θ(log n), and that adding sparsity inequalities reduces the integrality gap to 10.Our main result is a primal-dual algorithm that yields a 47/7 ≈ 6.71-approximation for ECT on node-weighted planar graphs, and an integrality gap of the same value for the standard LP relaxation on node-weighted planar graphs.

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