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Weighted Book Thickness

Authors Henry Förster , Michael Hoffmann , Stephen Kobourov , Maria Eleni Pavlidi , Alexandra Weinberger , Johannes Zink

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LIPIcs.GD.2026.27.pdf
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ACM Subject Classification
  • Mathematics of computing → Combinatorics
  • Mathematics of computing → Graph theory
  • Human-centered computing → Graph drawings
Keywords
  • book thickness
  • linear layouts
  • 2-trees

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Abstract

We introduce and study the weighted book thickness of graphs. A k-page book embedding of a graph G = (V,E) is defined by a spanning cycle C for V (which does not need to be part of G) and a partition E = ⋃_{i=1}^k E_i such that E∩ C ⊆ E₁ and each graph G_i = (V,E_i∪ C), for 1 ≤ i ≤ k, is outerplane with outer cycle C. If e ∈ E_i, we say that e appears on Page i. The classical book thickness of a graph G is the minimum k such that there exists a k-page book embedding of G, that is, the minimum (over all book embeddings of G) achievable maximum page an edge appears on. In contrast, the weighted book thickness is the minimum achievable average page an edge appears on. The embeddings that realize weighted book thickness can differ from those that realize (classical) book thickness. We show that, although every planar graph on at most nine vertices admits a 2-page book embedding realizing its weighted book thickness, already for ten vertices, there is a planar graph for which every realization of its weighted book thickness needs more pages than its book thickness. We prove that there even exists a 2-tree whose weighted book thickness cannot be realized on two pages. On the positive side, we show that for every graph of pathwidth at most two, the weighted book thickness can always be realized by a 2-page book embedding and such an embedding can be found in linear time. Moreover, we prove that it is NP-complete to decide if the weighted book thickness is at most k, for some given integer k.

Cite As Get BibTex

Henry Förster, Michael Hoffmann, Stephen Kobourov, Maria Eleni Pavlidi, Alexandra Weinberger, and Johannes Zink. Weighted Book Thickness. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 27:1-27:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.27

Author Details

Henry Förster
  • John Cabot University, Rome, Italy
  • Technical University of Munich, Heilbronn, Germany
Michael Hoffmann
  • Department of Computer Science, ETH Zürich, Switzerland
Stephen Kobourov
  • Technical University of Munich, Heilbronn, Germany
Maria Eleni Pavlidi
  • University of Ioannina, Greece
Alexandra Weinberger
  • FernUniversität in Hagen, Germany
Johannes Zink
  • Technische Universität München, Germany

Funding

  • Pavlidi, Maria Eleni: Maria Eleni Pavlidi was supported by HFRI Grant No: 26320.

Acknowledgements

This work was initiated at the SWGD'24 workshop.

Supplementary Materials

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