| Title: | A dichotomy for 1-planarity with restricted crossing types parameterized by treewidth |
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| Authors: | ID Cabello, Sergio (Author)
ID Dobler, Alexander (Author)
ID Fijavž, Gašper (Author)
ID Hamm, Thekla (Author)
ID Wagner, Mirko H. (Author) |
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| Files: |  PDF - Presentation file, download (843,13 KB)
MD5: 99F8A2DEAC75183A893C2DA5D3178B14
 URL - Source URL, visit https://drops.dagstuhl.de/entities/document/10.4230/LIPIcs.ISAAC.2025.16
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| Language: | English |
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| Typology: | 1.08 - Published Scientific Conference Contribution |
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| Organization: |  IMFM - Institute of Mathematics, Physics, and Mechanics
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| Abstract: | A drawing of a graph is 1-planar if each edge participates in at most one crossing and adjacent edges do not cross. Up to symmetry, each crossing in a 1-planar drawing belongs to one out of six possible crossing types, where a type characterizes the subgraph induced by the four vertices of the crossing edges. Each of the 63 possible nonempty subsets ${\mathcal S}$ of crossing types gives a recognition problem: does a given graph admit an ${\mathcal S}$-restricted drawing, that is, a 1-planar drawing where the crossing type of each crossing is in ${\mathcal S}$? We show that there is a set ${\mathcal S}_{\rm bad}$ with three crossing types and the following properties: (i) If ${\mathcal S}$ contains no crossing type from ${\mathcal S}_{\rm bad}$, then the recognition of graphs that admit an ${\mathcal S}$-restricted drawing is fixed-parameter tractable with respect to the treewidth of the input graph. (ii) If ${\mathcal S}$ contains any crossing type from ${\mathcal S}_{\rm bad}$, then it is NP-hard to decide whether a graph has an ${\mathcal S}$-restricted drawing, even when considering graphs of constant pathwidth. We also extend this characterization of crossing types to 1-planar straight-line drawings and show the same complexity behaviour parameterized by treewidth. |
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| Keywords: | 1-planar, crossing type, treewidth, pathwidth |
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| Publication status: | Published |
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| Publication version: | Version of Record |
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| Year of publishing: | 2025 |
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| Number of pages: | 15 str. |
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| PID: | 20.500.12556/DiRROS-25062  |
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| UDC: | 004.42:519.17 |
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| DOI: | 10.4230/LIPIcs.ISAAC.2025.16 |
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| COBISS.SI-ID: | 263974915 |
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| Note: |
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| Publication date in DiRROS: | 08.01.2026 |
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| Views: | 495 |
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| Downloads: | 249 |
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