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Burning Numbers of Barbells

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Abstract

Motivated by a discrete-time process intended to measure the speed of the spread of contagion in a graph, the burning number b(G) of a graph G, is defined as the smallest integer k for which there are vertices x1,…,xk such that for every vertex u of G, there exists i ∈ {1,…,k} with dG(u, xi) ≤ k − i, and dG(xi, xj) ≥ j − i for any 1 ≤ i < j ≤ k. The graph burning problem has been shown to be NP-complete even for some acyclic graphs with maximum degree three. In this paper, we determine the burning numbers of all short barbells and long barbells, respectively.

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References

  1. Alon, N., Prałat, P., Wormald, N. Cleaning regular graphs with brushes. SIAM Journal on Discrete Math., 23: 233–250 (2008)

    Article  MathSciNet  Google Scholar 

  2. Balogh, J., Bollobás, B., Morris, R. Graph bootstrap percolation. Random Struct. Alg., 41: 413–440 (2012)

    Article  MathSciNet  Google Scholar 

  3. Barghi, A., Winkler, P. Firefighting on a random geometric graph. Random Struct. Alg., 46: 466–477 (2015)

    Article  MathSciNet  Google Scholar 

  4. Bessy, S., Bonato, A., Janssen, J., Rautenbach, D., Roshanbin, E. Burning a graph is hard. Discrete Appl. Math., 232: 73–87 (2017)

    Article  MathSciNet  Google Scholar 

  5. Bessy, S., Bonato, A., Janssen, J., Rautenbach, D., Roshanbin, E. Bounds on the burning number. Discrete Appl. Math., 235: 16–22 (2018)

    Article  MathSciNet  Google Scholar 

  6. Bonato, A. A survey of graph burning. Contrib. Discrete Math., 16: 185–197 (2021)

    MathSciNet  Google Scholar 

  7. Bonato, A., Janssen, J., Roshanbin, E. Burning a graph as a model of social contagion. Lecture Notes in Comput. Sci., 8882: 13–22 (2014)

    Article  MathSciNet  Google Scholar 

  8. Bonato, A., Janssen, J., Roshanbin, E. How to burn a graph. Internet Math., 1–2: 85–100 (2016)

    Article  MathSciNet  Google Scholar 

  9. Bonato, A., Lidbetter, T. Bounds on the burning numbers of spiders and path-forests. Theor. Comput. Sci., 794: 12–19 (2019)

    Article  MathSciNet  Google Scholar 

  10. Bonato, A., Nowakowski, R.J. The game of cops and robbers on graphs, Vol. 61. Amer. Math. Soc., Providence, Rhode Island, 2011

    Google Scholar 

  11. Bonato, A., Prałat, P. Graph searching games and probabilistic methods. CRC Press, 2017

  12. Finbow, S., MacGillivray, G. The Firefighter problem: a survey of results, directions and questions. Australasian Journal of Comb., 43: 57–77 (2009)

    MathSciNet  Google Scholar 

  13. Fitzpatrick, S.L., Wilm, L. Burning circulant graphs. arXiv:1706.03106

  14. Land, M.R., Lu, L.Y. An upper bound on the burning number of graphs, In Algorithms and Models for the web graph, 1–8. Lecture Notes in Computer Science, Vol.10088. Springer, Cham, 2016

    Google Scholar 

  15. Liu, H.Q., Zhang R.T., Hu, X.L. Burning number of theta graphs. Appl. Math. Comput., 361: 246–257 (2019)

    MathSciNet  Google Scholar 

  16. Liu, H.Q., Hu, X.J., Hu, X.L. Burning numbers of path forests and spiders. Bull. Malays. Math. Sci. Soc., 44: 661–681 (2021)

    Article  MathSciNet  Google Scholar 

  17. Liu, H.Q., Hu, X.J., Hu, X.L. Burning number of caterpillars. Disc. Appl. Math., 284: 332–340 (2020)

    Article  MathSciNet  Google Scholar 

  18. Mitsche, D., Prałat, P., Roshanbin, E. Burning graphs: a probabilistic perspective. Graphs Comb., 33: 449–471 (2017)

    Article  MathSciNet  Google Scholar 

  19. Mitsche, D., Prałat, P., Roshanbin, E. Burning number of graph products. Theor. Comput. Sci., 746: 124–135 (2018)

    Article  MathSciNet  Google Scholar 

  20. Sim, K.A., Tan, T.S., Wong, K.B. On the burning number of generalized Petersen graphs. Bull. Malays. Math. Sci. Soc., 41: 1657–1670 (2018)

    Article  MathSciNet  Google Scholar 

  21. Zhang, R.T., Yu, Y.Y., Liu, H.Q. Burning numbers of t-unicyclic graphs. Bull. Malays. Math. Sci. Soc., 45, 417–430 (2022)

    Article  MathSciNet  Google Scholar 

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Authors and Affiliations

  1. Hubei Key Laboratory of Applied Mathematics & Faculty of Mathematics and Statistic, Hubei University, Wuhan, 430062, China

    Hui-qing Liu & Rui-ting Zhang

  2. Hubei Key Laboratory of Mathematical Sciences & School of Mathematics and Statistics, Central China Normal University, Wuhan, 430079, China

    Xiao-lan Hu

Authors
  1. Hui-qing Liu
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  2. Rui-ting Zhang
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  3. Xiao-lan Hu
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Correspondence to Xiao-lan Hu.

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The authors declare no conflict of interest.

Additional information

This paper is supported by the National Natural Science Foundation of China (Nos.11971158, 12371345, 11971196).

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Liu, Hq., Zhang, Rt. & Hu, Xl. Burning Numbers of Barbells. Acta Math. Appl. Sin. Engl. Ser. 40, 526–538 (2024). https://doi.org/10.1007/s10255-024-1113-8

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  • DOI: https://doi.org/10.1007/s10255-024-1113-8

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