research-article

Testing Planarity of Partially Embedded Graphs

Authors:

Patrizio Angelini,

Giuseppe Di Battista,

Fabrizio Frati,

Jan Kratochvíl,

Maurizio Patrignani,

Ignaz RutterAuthors Info & Claims

ACM Transactions on Algorithms (TALG), Volume 11, Issue 4

Article No.: 32, Pages 1 - 42

https://doi.org/10.1145/2629341

Published: 13 April 2015 Publication History

Abstract

We study the following problem: given a planar graph G and a planar drawing (embedding) of a subgraph of G, can such a drawing be extended to a planar drawing of the entire graph G? This problem fits the paradigm of extending a partial solution for a problem to a complete one, which has been studied before in many different settings. Unlike many cases, in which the presence of a partial solution in the input makes an otherwise easy problem hard, we show that the planarity question remains polynomial-time solvable. Our algorithm is based on several combinatorial lemmas, which show that the planarity of partially embedded graphs exhibits the ‘TONCAS’ behavior “the obvious necessary conditions for planarity are also sufficient.” These conditions are expressed in terms of the interplay between (1) the rotation system and containment relationships between cycles and (2) the decomposition of a graph into its connected, biconnected, and triconnected components. This implies that no dynamic programming is needed for a decision algorithm and that the elements of the decomposition can be processed independently.

Further, by equipping the components of the decomposition with suitable data structures and by carefully splitting the problem into simpler subproblems, we make our algorithm run in linear time.

Finally, we consider several generalizations of the problem, such as minimizing the number of edges of the partial embedding that need to be rerouted to extend it, and argue that they are NP-hard. We also apply our algorithm to the simultaneous graph drawing problem Simultaneous Embedding with Fixed Edges (Sefe). There we obtain a linear-time algorithm for the case that one of the input graphs or the common graph has a fixed planar embedding.

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Münch MRutter IStumpf P(2024)Partial and Simultaneous Transitive Orientations via Modular DecompositionsAlgorithmica10.1007/s00453-023-01188-y86:4(1263-1292)Online publication date: 1-Apr-2024
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Pinki PShekhawat K(2023)An annotated review on graph drawing and its applicationsAKCE International Journal of Graphs and Combinatorics10.1080/09728600.2023.2218459(1-24)Online publication date: 11-Jul-2023
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Index Terms

Testing Planarity of Partially Embedded Graphs
1. Mathematics of computing
  1. Discrete mathematics
    1. Graph theory

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cover image ACM Transactions on Algorithms

ACM Transactions on Algorithms Volume 11, Issue 4

June 2015

302 pages

ISSN:1549-6325

EISSN:1549-6333

DOI:10.1145/2756876

Editor:
Aravind Srinivasan
University of Maryland, USA

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Copyright © 2015 ACM.

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Publication History

Published: 13 April 2015

Accepted: 01 April 2014

Revised: 01 April 2014

Received: 01 March 2013

Published in TALG Volume 11, Issue 4

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Funding Sources

Australian Research Council
MIUR project AMANDA (Algorithmics for Massive and Networked Data) protocol 2012C4E3KT_001
ESF EuroGIGA GraDR as Czech Research
fellowship within the postdoctoral program of the German Academic Exchange Service (DAAD)

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Cited By

Brückner GRutter IStumpf P(2024)Extending Partial Representations of Circle Graphs in Near-Linear TimeAlgorithmica10.1007/s00453-024-01216-586:7(2152-2173)Online publication date: 26-Mar-2024
https://dl.acm.org/doi/10.1007/s00453-024-01216-5
Münch MRutter IStumpf P(2024)Partial and Simultaneous Transitive Orientations via Modular DecompositionsAlgorithmica10.1007/s00453-023-01188-y86:4(1263-1292)Online publication date: 1-Apr-2024
https://dl.acm.org/doi/10.1007/s00453-023-01188-y
Pinki PShekhawat K(2023)An annotated review on graph drawing and its applicationsAKCE International Journal of Graphs and Combinatorics10.1080/09728600.2023.2218459(1-24)Online publication date: 11-Jul-2023
https://doi.org/10.1080/09728600.2023.2218459
Arroyo AKlute FParada IVogtenhuber BSeidel RWiedera T(2022)Inserting One Edge into a Simple Drawing is HardDiscrete & Computational Geometry10.1007/s00454-022-00394-969:3(745-770)Online publication date: 13-Aug-2022
https://doi.org/10.1007/s00454-022-00394-9
Nöllenburg MSorge MTerziadis SVilledieu AWu HWulms J(2022)Planarizing Graphs and Their Drawings by Vertex SplittingGraph Drawing and Network Visualization10.1007/978-3-031-22203-0_17(232-246)Online publication date: 13-Sep-2022
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Fiala JRutter IStumpf PZeman P(2022)Extending Partial Representations of Circular-Arc GraphsGraph-Theoretic Concepts in Computer Science10.1007/978-3-031-15914-5_17(230-243)Online publication date: 1-Oct-2022
https://doi.org/10.1007/978-3-031-15914-5_17
Liotta GRutter ITappini A(2021)Simultaneous FPQ-ordering and hybrid planarity testingTheoretical Computer Science10.1016/j.tcs.2021.05.012874(59-79)Online publication date: Jun-2021
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Angelini PKindermann PLöffler ASchlipf LSymvonis A(2021)One-Bend Drawings of Outerplanar Graphs Inside Simple PolygonsGraph Drawing and Network Visualization10.1007/978-3-030-92931-2_13(184-192)Online publication date: 14-Sep-2021
https://dl.acm.org/doi/10.1007/978-3-030-92931-2_13
Da Lozzo GDi Battista GFrati F(2020)Extending upward planar graph drawingsComputational Geometry10.1016/j.comgeo.2020.10166891(101668)Online publication date: Dec-2020
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Rutter I(2020)Simultaneous EmbeddingBeyond Planar Graphs10.1007/978-981-15-6533-5_13(237-265)Online publication date: 1-Oct-2020
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