A 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.

翻译：若一个图允许每条边至多被交叉一次的直线段绘制，则称其为几何1-平面图。本文首次系统研究了判定几何1-平面图的参数化复杂度。通过大幅扩展Bannister、Cabello与Eppstein的技术，并结合Thomassen关于可直线化的1-平面嵌入之刻画，我们证明了当以树深（treedepth）为参数时，该问题是固定参数可解的。此外，我们获得了以反馈边数$\ell$为参数的几何1-平面性问题的核。作为副产品，我们将同一参数化下1-平面性与$k$-平面性已知的最佳核大小从$O((3\ell)!)$改进至$O(\ell \cdot 8^{\ell})$。我们的方法自然推广至几何$k$-平面性，从而在相同参数化下得到核化结果，尽管核尺寸更大。作为这些结果的补充，我们给出了匹配的下界：即使对于路径宽度、反馈顶点数与带宽均有界的图，几何1-平面性判定问题仍然是\NP-完全的。