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Extending Biconnected Straight-Line Planar Drawings

Authors Giordano Andreola , Susanna Caroppo , Giordano Da Lozzo , Marco D'Elia , Giuseppe Di Battista , Fabrizio Frati , Fabrizio Grosso , Maurizio Patrignani

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LIPIcs.GD.2026.4.pdf
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Subject Classification

ACM Subject Classification
  • Theory of computation → Computational geometry
  • Theory of computation → Design and analysis of algorithms
  • Mathematics of computing → Graph algorithms
Keywords
  • Planar Graphs
  • Straight-line Drawings
  • Extension Problems
  • FPT
  • Fixed Embedding

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Abstract

The Partial Drawing Extensibility problem, for short PDE, takes as input a triple ⟨ G,H,Γ_H⟩, where G is a planar graph, H is a subgraph of G, and Γ_H is a straight-line planar drawing of H, and asks whether Γ_H can be extended to a straight-line planar drawing of G. Patrignani [Int. J. Found. Comput. Sci. (2006)] proved that the PDE problem is NP-hard, exploiting instances in which H is highly disconnected. In this paper, we study the PDE problem under the requirement that the initial partial drawing Γ_H is biconnected. We show that PDE remains NP-hard even for instances in which H is a biconnected graph with faces of bounded size, G is subcubic, and the part of G that is not in H consists of length-2 paths. The complexity of PDE remains however open when H is connected (or even biconnected) if G has a fixed embedding. In this setting both a polynomial-time algorithm or an NP-hardness proof seem to be elusive targets. As a step towards tackling this problem, we study instances of PDE in which H is biconnected, G has a fixed embedding, and the rest of the graph consists of p length-2 paths, and present an O(p² n)-time algorithm, a result in sharp contrast with the NP-hardness of the variable embedding setting. Moreover, with an approach based on the Existential Theory of the Reals, we show that, if H is biconnected, the problem is FPT parameterized by the vertex cover number of G, both in a fixed and in a variable embedding setting.

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Giordano Andreola, Susanna Caroppo, Giordano Da Lozzo, Marco D'Elia, Giuseppe Di Battista, Fabrizio Frati, Fabrizio Grosso, and Maurizio Patrignani. Extending Biconnected Straight-Line Planar Drawings. In 34th International Symposium on Graph Drawing and Network Visualization (GD 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 396, pp. 4:1-4:21, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.GD.2026.4

Author Details

Giordano Andreola
  • Roma Tre University, Rome, Italy
Susanna Caroppo
  • Roma Tre University, Rome, Italy
Giordano Da Lozzo
  • Roma Tre University, Rome, Italy
Marco D'Elia
  • Roma Tre University, Rome, Italy
Giuseppe Di Battista
  • Roma Tre University, Rome, Italy
Fabrizio Frati
  • Roma Tre University, Italy
Fabrizio Grosso
  • CeDiPa - University of Perugia, Italy
Maurizio Patrignani
  • Roma Tre University, Rome, Italy

Funding

  • Grosso, Fabrizio: The seventh author was supported by Ce.Di.Pa. - PNC Programma unitario di interventi per le aree del terremoto del 2009-2016 - Linea di intervento 1 sub-misura B4 - "Centri di ricerca per l'innovazione" CUP J37G22000140001.

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