Approximate min-max relations on plane graphs

Let G be a plane graph, let τ(G) (resp. τ′(G)) be the minimum number of vertices (resp. edges) that meet all cycles of G, and let ν(G) (resp. ν′(G)) be the maximum number of vertex-disjoint (resp. edge-disjoint) cycles in G. In this note we show that τ(G)≤3ν(G) and τ′(G)≤4ν′(G)?1; our proofs are constructive, which yield polynomial-time algorithms for finding corresponding objects with the desired properties.