Abstract
The burning number is a recently introduced graph parameter indicating the spreading speed of content in a graph through its edges. While the conjectured upper bound on the necessary number of time steps until all vertices are reached is proven for some specific graph classes, it remains open for trees in general. We present two different proofs for ordinary caterpillars and prove the conjecture for a generalised version of caterpillars and for trees with a sufficient number of legs. Furthermore, determining the burning number for spider graphs, trees with maximum degree three and path-forests is known to be \(\mathcal {N}\mathcal {P}\)-complete; however, we show that the complexity is already inherent in caterpillars with maximum degree three.
Keywords
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
References
Bessy, S., Bonato, A., Janssen, J., Rautenbach, D., Roshanbin, E.: Burning a graph is hard. Discrete Appl. Math. 232, 73–87 (2017)
Bonato, A., Janssen, J., Roshanbin, E.: How to burn a graph. Internet Math. 12(1–2), 85–100 (2016)
Bonato, A., Lidbetter, T.: Bounds on the burning numbers of spiders and path-forests. Theor. Comput. Sci. 794, 12–19 (2019)
Harary, F., Schwenk, A.J.: The number of caterpillars. Discrete Math. 6(4), 359–365 (1973)
Hulett, H., Will, T.G., Woeginger, G.J.: Multigraph realizations of degree sequences: maximization is easy, minimization is hard. Oper. Res. Lett. 36(5), 594–596 (2008)
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2021 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this chapter
Cite this chapter
Hiller, M., Koster, A.C.A., Triesch, E. (2021). On the Burning Number of p-Caterpillars. In: Gentile, C., Stecca, G., Ventura, P. (eds) Graphs and Combinatorial Optimization: from Theory to Applications. AIRO Springer Series, vol 5. Springer, Cham. https://doi.org/10.1007/978-3-030-63072-0_12
Download citation
DOI: https://doi.org/10.1007/978-3-030-63072-0_12
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-030-63071-3
Online ISBN: 978-3-030-63072-0
eBook Packages: Mathematics and StatisticsMathematics and Statistics (R0)