A Note on the Cycle Isolation Number of Graphs

set D of vertices in a graph G is a cycle isolating set of G if G-N[D] contains no cycle. The cycle isolation number of G , denoted by ι _c(G) , is the minimum cardinality of a cycle isolating set of G . In this paper, we prove that if G is a connected graph of size m that is not a C_3 , then ι _c(G) ≤m+1/5 , and we characterize the extremal graphs. Moreover, we conjecture that if G is a connected graph of size m that is not a C_g , then ι _c(G) ≤m+1/g+2 , where g is the girth of G .