Isolation of Cycles

Abstract

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.