On cycle packings and feedback vertex sets

For a graph G, let fvs and cp denote the minimum size of a feedback vertex set in G and the maximum size of a cycle packing in G, respectively. Kloks, Lee, and Liu conjectured that fvs(G) ≤ 2 cp(G) if G is planar. They proved a weaker inequality, replacing 2 by 5. We improve this, replacing 5 by 3, and verifying the resulting inequality for graphs embedded in surfaces of nonnegative Euler characteristic. We also generalize to arbitrary surfaces. We show that, if a graph G embeds in a surface of Euler characteristic c ≤ 0, then fvs(G) ≤ 3 cp(G) + 103(−c). Lastly, we consider what the best possible bound on fvs might be, and give some open problems.