Abstract
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 \(\iota _c(G)\), 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 \(\iota _c(G)\le \frac{n}{4}\). 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 \(\iota _c(G)\le \frac{m+1}{5}\). Moreover, we characterize all connected graphs attaining this bound.
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The authors would like to thank the anonymous referees for their careful reading and valuable suggestions which have improved the presentation of this paper.
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This work was partially supported by the National Natural Science Foundation of China (No. 12171239).
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Cui, Q., Zhang, J. A Sharp Upper Bound on the Cycle Isolation Number of Graphs. Graphs and Combinatorics 39, 117 (2023). https://doi.org/10.1007/s00373-023-02717-w
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DOI: https://doi.org/10.1007/s00373-023-02717-w