Abstract
Half graphs and their variants, such as ladders, semi-ladders and co-matchings, are combinatorial objects that encode total orders in graphs. Works by Adler and Adler (Eur. J. Comb.; 2014) and Fabiański et al. (STACS; 2019) prove that in the powers of sparse graphs, one cannot find arbitrarily large objects of this kind. We provide nearly tight asymptotic lower and upper bounds on the maximum order of half graphs, parameterized on the power, in the following classes of sparse graphs: planar graphs, graphs with bounded maximum degree, graphs with bounded pathwidth or treewidth, and graphs excluding a fixed clique as a minor. As an essential part of this work, we prove a fully polynomial bound on the neighborhood complexity in planar graphs.
The full version of this work is available on arXiv [9].
This work is a part of project TOTAL that has received funding from the European Research Council (ERC) under the European Union’s Horizon 2020 research and innovation programme (grant agreement No. 677651).
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References
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Acknowledgements
We would like to thank: Michał Pilipczuk for pointing us to the problem and the proofreading of the full version of this work; Piotr Micek for reviewing the full version of the work; and an anonymous reviewer for their suggestions and comments.
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Sokołowski, M. (2021). Bounds on Half Graph Orders in Powers of Sparse Graphs. In: Nešetřil, J., Perarnau, G., Rué, J., Serra, O. (eds) Extended Abstracts EuroComb 2021. Trends in Mathematics(), vol 14. Birkhäuser, Cham. https://doi.org/10.1007/978-3-030-83823-2_39
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DOI: https://doi.org/10.1007/978-3-030-83823-2_39
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