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Extending Partial 1-Planar Drawings

Authors Eduard Eiben , Robert Ganian , Thekla Hamm, Fabian Klute , Martin Nöllenburg

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Author Details

Eduard Eiben
  • Department of Computer Science, Royal Holloway, University of London, Egham, United Kingdom
Robert Ganian
  • Algorithms and Complexity Group, TU Wien, Austria
Thekla Hamm
  • Algorithms and Complexity Group, TU Wien, Austria
Fabian Klute
  • Algorithms and Complexity Group, TU Wien, Austria
Martin Nöllenburg
  • Algorithms and Complexity Group, TU Wien, Austria

Funding

  • Ganian, Robert: Robert Ganian acknowledges support by the Austrian Science Fund (FWF, project P31336).
  • Hamm, Thekla: Thekla Hamm acknowledges support by the Austrian Science Fund (FWF, projects P31336 and W1255-N23).

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Eduard Eiben, Robert Ganian, Thekla Hamm, Fabian Klute, and Martin Nöllenburg. Extending Partial 1-Planar Drawings. In 47th International Colloquium on Automata, Languages, and Programming (ICALP 2020). Leibniz International Proceedings in Informatics (LIPIcs), Volume 168, pp. 43:1-43:19, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2020) https://doi.org/10.4230/LIPIcs.ICALP.2020.43

Abstract

Algorithmic extension problems of partial graph representations such as planar graph drawings or geometric intersection representations are of growing interest in topological graph theory and graph drawing. In such an extension problem, we are given a tuple (G,H,ℋ) consisting of a graph G, a connected subgraph H of G and a drawing ℋ of H, and the task is to extend ℋ into a drawing of G while maintaining some desired property of the drawing, such as planarity.
In this paper we study the problem of extending partial 1-planar drawings, which are drawings in the plane that allow each edge to have at most one crossing. In addition we consider the subclass of IC-planar drawings, which are 1-planar drawings with independent crossings. Recognizing 1-planar graphs as well as IC-planar graphs is NP-complete and the NP-completeness easily carries over to the extension problem. Therefore, our focus lies on establishing the tractability of such extension problems in a weaker sense than polynomial-time tractability. Here, we show that both problems are fixed-parameter tractable when parameterized by the number of edges missing from H, i.e., the edge deletion distance between H and G. The second part of the paper then turns to a more powerful parameterization which is based on measuring the vertex+edge deletion distance between the partial and complete drawing, i.e., the minimum number of vertices and edges that need to be deleted to obtain H from G.

Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Computational geometry
Keywords
  • Extension problems
  • 1-planarity
  • parameterized algorithms

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