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The Page Number of Monotone Directed Acyclic Outerplanar Graphs Is Four or Five

Authors Jawaherul Md. Alam , Michael A. Bekos , Martin Gronemann , Michael Kaufmann

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Graph theory
Keywords
  • Book embeddings
  • page number
  • directed outerplanar graphs

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Abstract

A k-page book embedding of a directed acyclic graph consists of a topological order of its vertices and a k-coloring of its edges, such that no two edges of the same color cross, that is, their endpoints do not alternate in the order. The minimum value of k for which such an embedding exists is referred to as the page number of the graph. In contrast to general directed acyclic planar graphs, which may have unbounded page number [SIAM J. Comput. 28(5), 1999], it was recently shown that directed acyclic outerplanar graphs have bounded page number. In particular, Jungeblut, Merker and Ueckerdt provided an upper bound of 24,776 on their page number [FOCS 2023: 1937-1952]. 
In this work, we focus on so-called monotone directed acyclic outerplanar graphs. Starting from a single edge, these graphs are constructed by iteratively connecting a new vertex to the endpoints of an existing edge on the outer face using either two incoming or two outgoing edges incident to it. These graphs have twist-number 4 [GD 2023: 135-151] (i.e., they admit a topological order in which no more than four edges pairwise cross), a property, which was leveraged by Jungeblut, Merker and Ueckerdt to show that their page number is at most 128. We lower this upper bound to 5 and we also provide a lower bound of 4. A notable consequence of our result is a significant improvement of the upper bound on the page number of general directed outerplanar graphs from 24,776 to 1,160.

Cite As Get BibTex

Jawaherul Md. Alam, Michael A. Bekos, Martin Gronemann, and Michael Kaufmann. The Page Number of Monotone Directed Acyclic Outerplanar Graphs Is Four or Five. In 33rd International Symposium on Graph Drawing and Network Visualization (GD 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 357, pp. 9:1-9:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.GD.2025.9

Author Details

Jawaherul Md. Alam
  • Amazon Inc., Tempe, AZ, USA
Michael A. Bekos
  • Department of Mathematics, University of Ioannina, Greece
Martin Gronemann
  • Independent Researcher, Dortmund, Germany
Michael Kaufmann
  • Institut für Informatik, Universität Tübingen, Germany

Funding

The work of M.B. has been supported by HFRI Grant No: 26320.

Acknowledgements

We would like to thank the anonymous reviewers of this submission for valuable comments; in particular, for bringing to our attention the details of Remark 5.

References

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