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Parameterized Algorithms for Fixed-Order Book Drawing with Bounded Number of Crossings per Edge

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Combinatorial Optimization and Applications (COCOA 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12577))

Abstract

Given a graph \(G=(V,E)\) and a fixed linear order of V, the problem fixed-order book drawing with bounded number of crossings per edge asks whether there is a k-page book drawing of G such that the maximum number of crossings per edge is upper-bounded by an integer b. This problem was posed by Bhore et al. (GD 2019; J. Graph Algorithms Appl. 2020) and thought to be interesting for further investigation. In this paper, we study the fixed-parameter tractable algorithms for this problem. More precisely, we show that this problem parameterized by both the bounded number b of crossings per edge and the vertex cover number \(\tau \) of the graph admits an algorithm running in time \((b+2)^{O(\tau ^3)}\cdotp |V|\), and this problem parameterized by both the bounded number b of crossings per edge and the pathwidth \(\kappa \) of the vertex ordering admits an algorithm running in time \((b+2)^{O(\kappa ^2)}\cdotp |V|\). Our results provide a specifical answer to Bhore et al.’s question.

This research was supported in part by the National Natural Science Foundation of China under Grants (No. 61572190, 61972423), and Hunan Provincial Science and Technology Program Foundations (No. 2018TP1018, 2018RS3065).

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Acknowledgements

The authors thank the anonymous referees for their valuable comments and suggestions.

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Authors and Affiliations

  1. Hunan Provincial Key Laboratory of Intelligent Computing and Language Information Processing, Hunan Normal University, Changsha, 410081, People’s Republic of China

    Yunlong Liu, Jie Chen & Jingui Huang

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  1. Yunlong Liu
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  2. Jie Chen
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  3. Jingui Huang
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Correspondence to Jingui Huang .

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Editors and Affiliations

  1. The University of Texas at Dallas, Richardson, TX, USA

    Weili Wu

  2. Xiamen University, Xiamen, China

    Zhongnan Zhang

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Liu, Y., Chen, J., Huang, J. (2020). Parameterized Algorithms for Fixed-Order Book Drawing with Bounded Number of Crossings per Edge. In: Wu, W., Zhang, Z. (eds) Combinatorial Optimization and Applications. COCOA 2020. Lecture Notes in Computer Science(), vol 12577. Springer, Cham. https://doi.org/10.1007/978-3-030-64843-5_38

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  • DOI: https://doi.org/10.1007/978-3-030-64843-5_38

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