A Sharp Upper Bound on the Cycle Isolation Number of Graphs.

For any graph G, a subset S of vertices of G is said to be a cycle isolating set of G if G - N G [ S ] contains no cycle, where N G [ S ] is the closed neighborhood of S. The cycle isolation number of G, denoted by ι c (G) , is the minimum cardinality of a cycle isolating set of G. Recently, Borg (2020) showed that if G is a connected n-vertex graph that is not isomorphic to C 3 , then ι c (G) ≤ n 4 . In this paper, we present a sharp upper bound on the cycle isolation number of a connected graph in terms of its number of edges. We prove that if G is a connected m-edge graph that is not isomorphic to C 3 , then ι c (G) ≤ m + 1 5 . Moreover, we characterize all connected graphs attaining this bound. [ABSTRACT FROM AUTHOR]