Is feedback vertex set problem solvable in polynomial time for 3-degree bounded graphs?
Li and Liu's algorithm is incorrect (see below).  Ueno et al.'s algorithm is correct, and a similar algorithm can be found in Furst et al. 1.  Both algorithms reduce the problem to the polynomial-solvable matroid parity problem [3].

Its reduction from VC ensures the NP-hardness for degree-6 bounded graph!  As VC is already NP-hard on cubic graphs. Speckenmeyer has claimed that his thesis [4] contains the proof of NP-hardness of FVS on planar graphs of maximum degree four, but it's very hard to find (I'll greatly appreciate if who have access to his thesis can send me a copy). Fortunately, a new proof of
the NP-hardness of degree-four bounded graphs can be found in 2:

Remarks on 2: 
 - In fact, he proved that the problem is APX-hard, but it is easy to verify
that his reductions are also valid for the proof of NP-hardness of the problem.
 - Its reduction does NOT apply to planar graphs. 


Merrick L. Furst, Jonathan L. Gross, and Lyle A. McGeoch, “Finding a maximum-genus graph imbedding,” Journal of the ACM, vol. 35, no. 3, pp. 523–534, 1988. 10.1145/44483.44485
Rizzi, R.: Weakly fundamental cycle bases are hard to find. Algorithmica 53(3), 402-424 (2009) 10.1007/s00453-007-9112-8
László Lovász, “The matroid matching problem,” in Algebraic Methods in Graph Theory, ser. Colloquia Mathematica Societatis János Bolyai, vol. 25, Szeged, Hungary, 1980, pp. 495–517.
Ewald Speckenmeyer, “Untersuchungen zum feedback vertex set problem in ungerichteten graphen,” PhD thesis, Universität-GH Paderborn, Reihe Informatik, Bericht, 1983.