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Isolation of Cycles

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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.

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Acknowledgements

The author wishes to thank the anonymous referees for checking the paper and providing constructive remarks.

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  1. Department of Mathematics, Faculty of Science, University of Malta, Msida, Malta

    Peter Borg

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  1. Peter Borg
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Correspondence to Peter Borg.

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Borg, P. Isolation of Cycles. Graphs and Combinatorics 36, 631–637 (2020). https://doi.org/10.1007/s00373-020-02143-2

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  • DOI: https://doi.org/10.1007/s00373-020-02143-2

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