Frames, $A$-paths and the Erd\H{o}s-P\'osa property

A key feature of Simonovits’ proof of the classic Erd\H{o}s-P'osa theorem is a simple subgraph of the host graph, a frame, that determines the outcome of the theorem. We transfer this frame technique to $A$-paths. With it we deduce a simple proof of Gallai’s theorem, although with a worse bound, and we verify the Erd\H{o}s-P'osa property for long and for even $A$-paths. We also show that even $A$-paths do not have the edge-Erd\H{o}s-P'osa property.