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Packing Trees into 1-Planar Graphs

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WALCOM: Algorithms and Computation (WALCOM 2020)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 12049))

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Abstract

We introduce and study the 1-planar packing problem: Given k graphs with n vertices \(G_1, \dots , G_k\), find a 1-planar graph that contains the given graphs as edge-disjoint spanning subgraphs. We mainly focus on the case when each \(G_i\) is a tree and \(k=3\). We prove that a triple consisting of three caterpillars or of two caterpillars and a path may not admit a 1-planar packing, while two paths and a special type of caterpillar always have one. We then study 1-planar packings with few crossings and prove that three paths (resp. cycles) admit a 1-planar packing with at most seven (resp. fourteen) crossings. We finally show that a quadruple consisting of three paths and a perfect matching with \(n \ge 12\) vertices admits a 1-planar packing, while such a packing does not exist if \(n \le 10\).

This work started at the Bertinoro Workshop on Graph Drawing 2019 and it is partially supported by: (i) MIUR grant 20174LF3T8, (ii) Dipartimento di Ingegneria - Università degli Studi di Perugia grants RICBASE2017WD and RICBA18WD, (iii) NFS grants CCF-1740858, CCF-1712119, DMS-1839274, DMS-1839307.

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Authors and Affiliations

  1. Department of Computer Science, University of Arizona, Tucson, USA

    Felice De Luca & Stephen Kobourov

  2. Dipartimento di Ingegneria, Università degli Studi di Perugia, Perugia, Italy

    Emilio Di Giacomo, Giuseppe Liotta & Alessandra Tappini

  3. School of Computer Science, University of Sydney, Sydney, Australia

    Seok-Hee Hong

  4. Department of Computer Science, Williams College, Williamstown, USA

    William Lenhart

  5. Department of Computer Science, University College Roosevelt, Middelburg, The Netherlands

    Henk Meijer

  6. Department of Computer Science, University of Lethbridge, Lethbridge, Canada

    Stephen Wismath

Authors
  1. Felice De Luca
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  2. Emilio Di Giacomo
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  3. Seok-Hee Hong
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  4. Stephen Kobourov
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  5. William Lenhart
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  6. Giuseppe Liotta
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  7. Henk Meijer
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  8. Alessandra Tappini
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  9. Stephen Wismath
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Corresponding author

Correspondence to Alessandra Tappini .

Editor information

Editors and Affiliations

  1. Department of CSE, BUET, Dhaka, Bangladesh

    M. Sohel Rahman

  2. Mathematical Informatics, University of Tokyo, Tokyo, Japan

    Kunihiko Sadakane

  3. School of Computing, National University of Singapore, Singapore, Singapore

    Wing-Kin Sung

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De Luca, F. et al. (2020). Packing Trees into 1-Planar Graphs. In: Rahman, M., Sadakane, K., Sung, WK. (eds) WALCOM: Algorithms and Computation. WALCOM 2020. Lecture Notes in Computer Science(), vol 12049. Springer, Cham. https://doi.org/10.1007/978-3-030-39881-1_8

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  • DOI: https://doi.org/10.1007/978-3-030-39881-1_8

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-030-39880-4

  • Online ISBN: 978-3-030-39881-1

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