For a graph , let denote the cardinality of a maximum cycle packing (collection of vertex disjoint cycles) and let denote the cardinality of a minimum feedback vertex set (set of vertices so that is acyclic).
In [KLL], the authors mention that there exists a family of nonplanar graphs for which , so no such result could hold for general graphs. They also point out that the conjecture is tight for wheels, and they prove it for the special case of outerplanar graphs.
*[KLL] Ton Kloks, Chuan-Min Lee, and Jiping Liu, New Algorithms for -Face Cover, -Feedback Vertex Set, and -Disjoint Cycles on Plane and Planar Graphs, in Proceedings of the 28th International Workshop on Graph-Theoretic Concepts in Computer Science (WG2002), LNCS 2573, pp. 282--295, 2002.
* indicates original appearance(s) of problem.