Abstract
Let G be an undirected graph. We say that G contains a ladder of length k if the \(2 \times (k+1)\) grid graph is an induced subgraph of G that is only connected to the rest of G via its four cornerpoints. We prove that if all the ladders contained in G are reduced to length 4, the treewidth remains unchanged (and that this bound is tight). Our result indicates that, when computing the treewidth of a graph, long ladders can simply be reduced, and that minimal forbidden minors for bounded treewidth graphs cannot contain long ladders. Our result also settles an open problem from algorithmic phylogenetics: the common chain reduction rule, used to simplify the comparison of two evolutionary trees, is treewidth-preserving in the display graph of the two trees.
R. Meuwese was supported by the Dutch Research Council (NWO) KLEIN 1 grant Deep kernelization for phylogenetic discordance, project number OCENW.KLEIN.305.
This is a preview of subscription content, log in via an institution to check access.
Access this chapter
Tax calculation will be finalised at checkout
Purchases are for personal use only
Notes
- 1.
In every bag of the decomposition vertices from \(H_1\) all receive the vertex label \(H_1\), and similarly for the other subsets \(H_2, L_1, L_2\).
References
Abu-Khzam, F.N., Lamm, S., Mnich, M., Noe, A., Schulz, C., Strash, D.: Recent advances in practical data reduction. In: Bast, H., Korzen, C., Meyer, U., Penschuck, M. (eds.) Algorithms for Big Data. LNCS, vol. 13201, pp. 97–133. Springer, Cham (2022). https://doi.org/10.1007/978-3-031-21534-6_6
Allen, B., Steel, M.: Subtree transfer operations and their induced metrics on evolutionary trees. Ann. Comb. 5, 1–15 (2001)
Blair, J., Peyton, B.: Graph Theory and Sparse Matrix Computation, chap. An Introduction to Chordal Graphs and Clique Trees. In: George, A., Gilbert, J.R., Liu, J.W.H. (eds.) Graph Theory and Sparse Matrix Computation. The IMA Volumes in Mathematics and its Applications, vol. 56, pp. 1–29. Springer, New York (1993). https://doi.org/10.1007/978-1-4613-8369-7_1
Bodlaender, H.L., van Antwerpen-de Fluiter, B.: Reduction algorithms for graphs of small treewidth. Inf. Comput. 167(2), 86–119 (2001)
Bodlaender, H.L., Fomin, F.V., Lokshtanov, D., Penninkx, E., Saurabh, S., Thilikos, D.M.: (meta) kernelization. J. ACM 63(5), 44:1–44:69 (2016)
Bryant, D., Lagergren, J.: Compatibility of unrooted phylogenetic trees is FPT. Theoret. Comput. Sci. 351(3), 296–302 (2006)
Bulteau, L., Weller, M.: Parameterized algorithms in bioinformatics: an overview. Algorithms 12(12), 256 (2019)
Chaplick, S., Kelk, S., Meuwese, R., Mihalak, M., Stamoulis, G.: Snakes and ladders: a treewidth story. arXiv preprint arXiv:2302.10662 (2023)
Fomin, F.V., Lokshtanov, D., Saurabh, S., Zehavi, M.: Kernelization: Theory of Parameterized Preprocessing. Cambridge University Press, Cambridge (2019)
van Iersel, L., Jones, M., Weller, M.: Embedding phylogenetic trees in networks of low treewidth. In: Chechik, S., Navarro, G., Rotenberg, E., Herman, G. (eds.) 30th Annual European Symposium on Algorithms, ESA 2022, September 5–9, 2022, Berlin/Potsdam, Germany. LIPIcs, vol. 244, pp. 69:1–69:14. Schloss Dagstuhl - Leibniz-Zentrum für Informatik (2022). arXiv preprint arXiv:2207.00574
Janssen, R., Jones, M., Kelk, S., Stamoulis, G., Wu, T.: Treewidth of display graphs: bounds, brambles and applications. J. Graph Algorithms Appl. 23(4), 715–743 (2019)
John, K.S.: The shape of phylogenetic treespace. Syst. Biol. 66(1), e83 (2017)
Kelk, S., van Iersel, L., Scornavacca, C., Weller, M.: Phylogenetic incongruence through the lens of monadic second order logic. J. Graph Algorithms Appl. 20(2), 189–215 (2016)
Kelk, S., Stamoulis, G., Wu, T.: Treewidth distance on phylogenetic trees. Theoret. Comput. Sci. 731, 99–117 (2018)
Sanders, D.: On linear recognition of tree-width at most four. SIAM J. Discret. Math. 9(1), 101–117 (1996)
Steel, M.: Phylogeny: Discrete and Random Processes in Evolution. SIAM (2016)
Acknowledgements
We thank Hans Bodlaender and Bart Jansen for insightful feedback. We also thank the members of our department for useful discussions.
Author information
Authors and Affiliations
Corresponding author
Editor information
Editors and Affiliations
Rights and permissions
Copyright information
© 2023 The Author(s), under exclusive license to Springer Nature Switzerland AG
About this paper
Cite this paper
Chaplick, S., Kelk, S., Meuwese, R., Mihalák, M., Stamoulis, G. (2023). Snakes and Ladders: A Treewidth Story. In: Paulusma, D., Ries, B. (eds) Graph-Theoretic Concepts in Computer Science. WG 2023. Lecture Notes in Computer Science, vol 14093. Springer, Cham. https://doi.org/10.1007/978-3-031-43380-1_14
Download citation
DOI: https://doi.org/10.1007/978-3-031-43380-1_14
Published:
Publisher Name: Springer, Cham
Print ISBN: 978-3-031-43379-5
Online ISBN: 978-3-031-43380-1
eBook Packages: Computer ScienceComputer Science (R0)