TY - GEN
T1 - Extending Orthogonal Planar Graph Drawings Is Fixed-Parameter Tractable
AU - Bhore, Sujoy
AU - Ganian, Robert
AU - Khazaliya, Liana
AU - Montecchiani, Fabrizio
AU - Nöllenburg, Martin
N1 - Publisher Copyright:
© Sujoy Bhore, Robert Ganian, Liana Khazaliya, Fabrizio Montecchiani, and Martin Nöllenburg; licensed under Creative Commons License CC-BY 4.0.
PY - 2023/6/1
Y1 - 2023/6/1
N2 - 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.
AB - 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.
KW - bend minimization
KW - extension problems
KW - orthogonal drawings
KW - parameterized complexity
UR - http://www.scopus.com/inward/record.url?scp=85163563053&partnerID=8YFLogxK
U2 - 10.4230/LIPIcs.SoCG.2023.18
DO - 10.4230/LIPIcs.SoCG.2023.18
M3 - Conference contribution
AN - SCOPUS:85163563053
T3 - Leibniz International Proceedings in Informatics, LIPIcs
BT - 39th International Symposium on Computational Geometry, SoCG 2023
A2 - Chambers, Erin W.
A2 - Gudmundsson, Joachim
PB - Schloss Dagstuhl- Leibniz-Zentrum fur Informatik GmbH, Dagstuhl Publishing
T2 - 39th International Symposium on Computational Geometry, SoCG 2023
Y2 - 12 June 2023 through 15 June 2023
ER -