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# Extending Orthogonal Planar Graph Drawings Is Fixed-Parameter Tractable

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LIPIcs.SoCG.2023.18.pdf
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## Cite As

Sujoy Bhore, Robert Ganian, Liana Khazaliya, Fabrizio Montecchiani, and Martin Nöllenburg. Extending Orthogonal Planar Graph Drawings Is Fixed-Parameter Tractable. In 39th International Symposium on Computational Geometry (SoCG 2023). Leibniz International Proceedings in Informatics (LIPIcs), Volume 258, pp. 18:1-18:16, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2023)
https://doi.org/10.4230/LIPIcs.SoCG.2023.18

## Abstract

The task of finding an extension to a given partial drawing of a graph while adhering to constraints on the representation has been extensively studied in the literature, with well-known results providing efficient algorithms for fundamental representations such as planar and beyond-planar topological drawings. In this paper, we consider the extension problem for bend-minimal orthogonal drawings of planar graphs, which is among the most fundamental geometric graph drawing representations. While the problem was known to be NP-hard, it is natural to consider the case where only a small part of the graph is still to be drawn. Here, we establish the fixed-parameter tractability of the problem when parameterized by the size of the missing subgraph. Our algorithm is based on multiple novel ingredients which intertwine geometric and combinatorial arguments. These include the identification of a new graph representation of bend-equivalent regions for vertex placement in the plane, establishing a bound on the treewidth of this auxiliary graph, and a global point-grid that allows us to discretize the possible placement of bends and vertices into locally bounded subgrids for each of the above regions.

## Subject Classification

##### ACM Subject Classification
• Theory of computation → Computational geometry
• Mathematics of computing → Graph algorithms
##### Keywords
• orthogonal drawings
• bend minimization
• extension problems
• parameterized complexity

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