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Dynamic Embeddings of Dynamic Single-Source Upward Planar Graphs

Authors Ivor van der Hoog , Irene Parada , Eva Rotenberg

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Author Details

Ivor van der Hoog
  • Technical University of Denmark, Lyngby, Denmark
Irene Parada
  • Department of Mathematics, Universitat Politècnica de Catalunya, Barcelona, Spain
Eva Rotenberg
  • Technical University of Denmark, Lyngby, Denmark

Funding

This research was supported by Independent Research Fund Denmark grant 2020-2023 (9131-00044B) "Dynamic Network Analysis".
  • van der Hoog, Ivor: This project has additionally received funding from the European Union’s Horizon 2020 research and innovation programme under the Marie Skłodowska-Curie grant agreement No 899987.
  • Parada, Irene: I. P. is a Serra Húnter Fellow. Partially supported by grant 2021UPC-MS-67392 funded by the Spanish Ministry of Universities and the European Union (NextGenerationEU) and by grant PID2019-104129GB-I00 funded by MICIU/AEI/10.13039/501100011033.
  • Rotenberg, Eva: This research was additionally supported by Carlsberg Foundation Young Researcher Fellowship CF21-0302 "Graph Algorithms with Geometric Applications".

Cite AsGet BibTex

Ivor van der Hoog, Irene Parada, and Eva Rotenberg. Dynamic Embeddings of Dynamic Single-Source Upward Planar Graphs. In 32nd Annual European Symposium on Algorithms (ESA 2024). Leibniz International Proceedings in Informatics (LIPIcs), Volume 308, pp. 70:1-70:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2024)
https://doi.org/10.4230/LIPIcs.ESA.2024.70

Abstract

A directed graph G is upward planar if it admits a planar embedding where each edge is y-monotone. Unlike planarity testing, upward planarity testing is NP-hard except in restricted cases, such as when the graph has the single-source property (i.e., each connected component has one source). In this paper, we present a dynamic data structure for maintaining an upward combinatorial embedding ℰ→(G) of a single-source upward planar graph subject to edge deletions, edge contractions, directed edge insertions across a face, and single-source-preserving vertex splits through specified corners (i.e., the gaps between pairs of consecutive edges that share a vertex and a face). We furthermore support changes to the embedding ℰ→(G) in the form of subgraph flips that mirror or slide the placement of a subgraph that is connected to the rest of the graph via at most two vertices. Updates that are incompatible with the current upward planar embedding are identified and rejected. All update operations are supported as long as the graph remains upward planar. In addition, we support queries that can tell whether two vertices can be connected with a directed edge while the graph remains single-source (we call these uplinkability queries). If a pair of vertices are not uplinkable, we facilitate one-flip-linkable queries: These point to a flip that makes them uplinkable, if any such flip exists. We dynamically maintain a linear-size data structure on G which supports incidence queries between a vertex and a face, and uplinkability queries for vertex pairs. We support all updates and queries in O(log² n) worst-case time.

Subject Classification

ACM Subject Classification
  • Theory of computation → Dynamic graph algorithms
  • Mathematics of computing → Graphs and surfaces
Keywords
  • dynamic graphs
  • data structures
  • computational geometry
  • graph drawing
  • graph algorithms
  • upward planarity

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