  In directed graphs, we investigate the problems of finding: 1) a minimum
feedback vertex set (also called the Feedback Vertex Set problem, or MFVS), 2)
a feedback vertex set inducing an acyclic graph (also called the Vertex
2-Coloring without Monochromatic Cycles problem, or Acyclic FVS) and 3) a
minimum feedback vertex set inducing an acyclic graph (Acyclic MFVS).
  We show that these problems are strongly related to (variants of) Monotone
3-SAT and Monotone NAE 3-SAT, where monotone means that all literals are in
positive form. As a consequence, we deduce several NP-completeness results on
restricted versions of these problems. In particular, we define the 2-Choice
version of an optimization problem to be its restriction where the optimum
value is known to be either D or D+1 for some integer D, and the problem is
reduced to decide which of D or D+1 is the optimum value. We show that the
2-Choice versions of MFVS, Acyclic MFVS, Min Ones Monotone 3-SAT and Min Ones
Monotone NAE 3-SAT are NP-complete. The two latter problems are the variants of
Monotone 3-SAT and respectively Monotone NAE 3-SAT requiring that the truth
assignment minimize the number of variables set to true.
  Finally, we propose two classes of directed graphs for which Acyclic FVS is
polynomially solvable, namely flow reducible graphs (for which MFVS is already
known to be polynomially solvable) and C1P-digraphs (defined by an adjacency
matrix with the Consecutive Ones Property).
