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Upward Book Embeddings of st-Graphs

Authors Carla Binucci, Giordano Da Lozzo, Emilio Di Giacomo, Walter Didimo, Tamara Mchedlidze, Maurizio Patrignani

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  • Details for the omitted and sketched proofs can be found in the full version of the paper [Carla Binucci et al., 2019], which is available at https://arxiv.org/abs/1903.07966.

Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Mathematics of computing → Graph algorithms
Keywords
  • Upward Book Embeddings
  • st-Graphs
  • SPQR-trees
  • Branchwidth
  • Sphere-cut Decomposition

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Abstract

We study k-page upward book embeddings (kUBEs) of st-graphs, that is, book embeddings of single-source single-sink directed acyclic graphs on k pages with the additional requirement that the vertices of the graph appear in a topological ordering along the spine of the book. We show that testing whether a graph admits a kUBE is NP-complete for k >= 3. A hardness result for this problem was previously known only for k = 6 [Heath and Pemmaraju, 1999]. Motivated by this negative result, we focus our attention on k=2. On the algorithmic side, we present polynomial-time algorithms for testing the existence of 2UBEs of planar st-graphs with branchwidth b and of plane st-graphs whose faces have a special structure. These algorithms run in O(f(b)* n+n^3) time and O(n) time, respectively, where f is a singly-exponential function on b. Moreover, on the combinatorial side, we present two notable families of plane st-graphs that always admit an embedding-preserving 2UBE.

Cite As Get BibTex

Carla Binucci, Giordano Da Lozzo, Emilio Di Giacomo, Walter Didimo, Tamara Mchedlidze, and Maurizio Patrignani. Upward Book Embeddings of st-Graphs. In 35th International Symposium on Computational Geometry (SoCG 2019). Leibniz International Proceedings in Informatics (LIPIcs), Volume 129, pp. 13:1-13:22, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2019) https://doi.org/10.4230/LIPIcs.SoCG.2019.13

Author Details

Carla Binucci
  • Università degli Studi di Perugia, Perugia, Italy
Giordano Da Lozzo
  • Roma Tre University, Rome, Italy
Emilio Di Giacomo
  • Università degli Studi di Perugia, Perugia, Italy
Walter Didimo
  • Università degli Studi di Perugia, Perugia, Italy
Tamara Mchedlidze
  • Karlsruhe Institute of Technology, Karlsruhe, Germany
Maurizio Patrignani
  • Roma Tre University, Rome, Italy

Funding

This work was supported in part by project "Algoritmi e sistemi di analisi visuale di reti complesse e di grandi dimensioni" - Ricerca di Base 2018, Dipartimento di Ingegneria dell'Università degli Studi di Perugia (Binucci, Di Giacomo, and Didimo), and in part by MIUR Project "MODE" under PRIN 20157EFM5C, by MIUR Project "AHeAD" under PRIN 20174LF3T8, by MIUR-DAAD JMP No 34120, by H2020-MSCA-RISE project 734922 - "CONNECT", and by Roma Tre University Azione 4 Project "GeoView" (Da Lozzo and Patrignani).

Acknowledgements

This research began at the Bertinoro Workshop on Graph Drawing 2018.

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