For any graph G, let \(\iota _{\mathrm{c}}(G)\) denote the size of a smallest set D of vertices of G such that the graph obtained from G by deleting the closed neighbourhood of D contains no cycle. We prove that if G is a connected n-vertex graph that is not a triangle, then \(\iota _{\mathrm{c}}(G) \le n/4\). We also show that the bound is sharp. Consequently, this settles a problem of Caro and Hansberg.

中文翻译：

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隔离周期

对于任何图G，令\（\ iota _ {\ mathrm {c}}（G）\）表示G顶点的最小集合D的大小，这样通过删除D的闭合邻域从G获得的图包含没有周期。我们证明，如果G是一个非三角形的连通n-顶点图，则\（\ iota _ {\ mathrm {c}}（G）\ le n / 4 \）。我们还表明边界是尖锐的。因此，这解决了卡洛和汉斯伯格的问题。