Packing cycles through prescribed vertices under modularity constraints

The well-known theorem of Erd?s–Pósa says that either a graph G has k disjoint cycles or there is a vertex set X   of order at most f(k)$f(k)$ for some function f   such that G?X$G?X$ is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization.