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  "path": "/abs/2602.09978v1",
  "publishedAt": "2026-02-11T01:00:00.000Z",
  "site": "https://arxiv.org",
  "tags": [
    "Alexander Firbas"
  ],
  "textContent": "**Authors:** Alexander Firbas\n\nA graph is geometric 1-planar if it admits a straight-line drawing where each edge is crossed at most once. We provide the first systematic study of the parameterized complexity of recognizing geometric 1-planar graphs. By substantially extending a technique of Bannister, Cabello, and Eppstein, combined with Thomassen's characterization of 1-planar embeddings that can be straightened, we show that the problem is fixed-parameter tractable when parameterized by treedepth. Furthermore, we obtain a kernel for Geometric 1-Planarity parameterized by the feedback edge number $\\ell$. As a by-product, we improve the best known kernel size of $O((3\\ell)!)$ for 1-Planarity and $k$-Planarity under the same parameterization to $O(\\ell \\cdot 8^{\\ell})$. Our approach naturally extends to Geometric $k$-Planarity, yielding a kernelization under the same parameterization, albeit with a larger kernel. Complementing these results, we provide matching lower bounds: Geometric 1-Planarity remains \\NP-complete even for graphs of bounded pathwidth, bounded feedback vertex number, and bounded bandwidth.",
  "title": "The Parameterized Complexity of Geometric 1-Planarity"
}