Fixed-parameter tractability for subset feedback set problems with parity constraints

The subset feedback set problem, which is a generalization of the well-known feedback vertex set problem, is that we are given an undirected graph G with a vertex subset S and a positive integer k, and the goal is to find a vertex set X of size at most k   such that G−XG−X has no S-cycle, where an S-cycle is a cycle having at least one vertex of S. It was recently shown that this problem is fixed parameter tractable, where k is the parameter. In this paper, we further generalize this problem to one with the parity constraints, and show the fixed parameter tractability:1.For a parameter k, there exists a fixed-parameter algorithm that either finds a vertex set X of size k   such that G−XG−X has no S-cycle of even length, or concludes that such a vertex set does not exist.2.For a parameter k, there exists a fixed-parameter algorithm that either finds a vertex set X of size k   such that G−XG−X has no S-cycle of odd length, or concludes that such a vertex set does not exist.