Destroying cycles in digraphs

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For a simple directed graph G with no directed triangles, let β(G) be the size of the smallest subset X ∈ E(G) such that G\X has no directed cycles, and let γ(G) denote the number of unordered pairs of nonadjacent vertices in G. Chudnovsky, Seymour, and Sullivan showed that β(G) ≤ γ(G), and conjectured that β(G) ≤ $\tfrac{{\gamma (G)}} {2}$. In this paper we prove that β(G)<0.88γ(G).