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Extending Upward Planar Graph Drawings

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Algorithms and Data Structures (WADS 2019)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 11646))

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Abstract

In this paper we study the computational complexity of the Upward Planarity Extension problem, which takes as input an upward planar drawing \(\varGamma _H\) of a subgraph H of a directed graph G and asks whether \(\varGamma _H\) can be extended to an upward planar drawing of G.

We show that the Upward Planarity Extension problem is NP-complete, even if G has a prescribed upward embedding, the vertex set of H coincides with the one of G, and H contains no edge. Conversely, we show that the Upward Planarity Extension problem can be solved in \(O(n \log n)\) time if G is an n-vertex upward planar st-graph. This result improves upon a known \(O(n^2)\)-time algorithm, which however applies to all n-vertex single-source upward planar graphs. We also show how to solve in polynomial time a surprisingly difficult version of the Upward Planarity Extension problem, in which the underlying graph of G is a path or a cycle, G has a prescribed upward embedding, H contains no edges, and no two vertices share the same y-coordinate in \(\varGamma _H\).

This research was supported in part by MIUR Project “MODE” under PRIN 20157EFM5C, by MIUR Project “AHeAD” under PRIN 20174LF3T8, by H2020-MSCA-RISE project 734922 – “CONNECT”, and by MIUR-DAAD JMP N\(^\circ \) 34120.

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Acknowledgments

Lemma 4 comes from a research session the third author had with Ignaz Rutter, to which our thanks go.

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Authors and Affiliations

  1. Roma Tre University, Rome, Italy

    Giordano Da Lozzo, Giuseppe Di Battista & Fabrizio Frati

Authors
  1. Giordano Da Lozzo
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  2. Giuseppe Di Battista
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  3. Fabrizio Frati
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Corresponding author

Correspondence to Giordano Da Lozzo .

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Editors and Affiliations

  1. Department of Computing Science, University of Alberta, Edmonton, AB, Canada

    Zachary Friggstad

  2. School of Computer Science, Carleton University, Manotick, ON, Canada

    Jörg-Rüdiger Sack

  3. Department of Computing Science, University of Alberta, Edmonton, AB, Canada

    Mohammad R Salavatipour

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Da Lozzo, G., Di Battista, G., Frati, F. (2019). Extending Upward Planar Graph Drawings. In: Friggstad, Z., Sack, JR., Salavatipour, M. (eds) Algorithms and Data Structures. WADS 2019. Lecture Notes in Computer Science(), vol 11646. Springer, Cham. https://doi.org/10.1007/978-3-030-24766-9_25

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  • DOI: https://doi.org/10.1007/978-3-030-24766-9_25

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