  Lov\'asz (1965) characterized graphs without two vertex-disjoint cycles,
which implies that such graphs have at most three vertices hitting all cycles.
In this paper, we ask whether such a small hitting set exists for $S$-cycles,
when a graph has no two vertex-disjoint $S$-cycles. For a graph $G$ and a
vertex set $S$ of $G$, an $S$-cycle is a cycle containing a vertex of $S$.
  We provide an example $G$ on $21$ vertices where $G$ has no two
vertex-disjoint $S$-cycles, but three vertices are not sufficient to hit all
$S$-cycles. On the other hand, we show that four vertices are enough to hit all
$S$-cycles whenever a graph has no two vertex-disjoint $S$-cycles.
