Packing cycles through prescribed vertices

The well-known theorem of Erdo{double acute accent}s and Pósa says that a graph G has either k vertex-disjoint cycles or a vertex set X of order at most f(k) such that G\X is a forest. Starting with this result, there are many results concerning packing and covering cycles in graph theory and combinatorial optimization. In this paper, we generalize Erdo{double acute accent}s-Pósa's result to cycles that are required to go through a set S of vertices. Given an integer k and a vertex subset S (possibly unbounded number of vertices) in a given graph G, we prove that either G has k vertex-disjoint cycles, each of which contains at least one vertex of S, or G has a vertex set X of order at most f(k)=40k²log₂k such that G\X has no cycle that intersects S.