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We prove that $G$ contains a set $X$ of vertices such that $G-X$ has no directed cycle, and $|X|\\le \\tfrac{5n-5}9$ if $g=4$, $|X|\\le \\tfrac{2n-5}4$ if $g=5$, and $|X|\\le \\tfrac{2n-6}{g}$ if $g\\ge 6$. 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