Packing cycles through prescribed vertices

The well-known theorem of Erdős 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. s paper, we generalize Erdős–Pósaʼs 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 ) = 40 k 2 log 2 k such that G ∖ X has no cycle that intersects S.