Isolation of Cycles

For any graph G, let iota c(G)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\iota _{\mathrm{c}}(G)$$\end{document} 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 c(G)<= n/4\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\iota _{\mathrm{c}}(G) \le n/4$$\end{document}. We also show that the bound is sharp. Consequently, this settles a problem of Caro and Hansberg.