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Two-Page Book Embeddings of 4-Planar Graphs

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Abstract

Back in the eighties, Heath [Algorithms for embedding graphs in books. PhD thesis, University of North Carolina, Chapel Hill, 1985] showed that every 3-planar graph is subhamiltonian and asked whether this result can be extended to a class of graphs of degree greater than three. In this paper we affirmatively answer this question for the class of 4-planar graphs. Our contribution consists of two algorithms: The first one is limited to triconnected graphs, but runs in linear time and uses existing methods for computing hamiltonian cycles in planar graphs. The second one, which solves the general case of the problem, is a quadratic-time algorithm based on the book embedding viewpoint of the problem.

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Notes

  1. The bridge-blocks of a connected graph G are the connected components formed by deleting all bridges of G. The bridge-blocks and the bridges of G have a natural tree structure, called bridge-block tree.

  2. Note that the existence of this pair of consecutive anchors of \(\overline{T}\) is implied by Lemma 7; since for each ancillary c of a labeled anchored tree \(\overline{T}\) there exist at least an anchor of \(\overline{T}\) with label smaller than that of c and at least another with label greater than that of c, there should be two consecutive ones with this property as well.

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Acknowledgments

Work on this problem began at Dagstuhl Seminar 13151. We thank the organizers, participants and Prof. Dr. M. Kaufmann.

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Authors and Affiliations

  1. Wilhelm-Schickhard-Institut für Informatik, Universität Tübingen, Tübingen, Germany

    Michael A. Bekos

  2. Institut für Informatik, Universität zu Köln, Cologne, Germany

    Martin Gronemann

  3. School of Applied Mathematical and Physical Sciences, NTUA, Athens, Greece

    Chrysanthi N. Raftopoulou

Authors
  1. Michael A. Bekos
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  2. Martin Gronemann
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  3. Chrysanthi N. Raftopoulou
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Corresponding author

Correspondence to Michael A. Bekos.

Additional information

This article is based on the preliminary version: [Michael A. Bekos, Martin Gronemann and Chrysanthi N. Raftopoulou: Two-Page Book Embeddings of 4-Planar Graphs In N. Portier and E. W. Mayr editors, Proc. of 31st Symposium on Theoretical Aspects of Computer Science (STACS2014), LIPIcs, pp. 137–148, 2014.].

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Bekos, M.A., Gronemann, M. & Raftopoulou, C.N. Two-Page Book Embeddings of 4-Planar Graphs. Algorithmica 75, 158–185 (2016). https://doi.org/10.1007/s00453-015-0016-8

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  • DOI: https://doi.org/10.1007/s00453-015-0016-8

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