In a Lombardi drawing of a graph the vertices are drawn as points and the edges are drawn as circular arcs connecting their respective endpoints. Additionally, all vertices have perfect angular resolution, i.e., all angles incident to a vertex $v$ have size $2\pi/\mathrm{deg}(v)$. We prove that it is $\exists\mathbb{R}$-complete to determine whether a given graph admits a Lombardi drawing respecting a fixed cyclic ordering of the incident edges around each vertex. In particular, this implies NP-hardness. While most previous work studied the (non-)existence of Lombardi drawings for different graph classes, our result is the first on the computational complexity of finding Lombardi drawings of general graphs.

翻译：在图论中，伦巴第绘制要求将顶点绘制为点，边绘制为连接其对应端点的圆弧。此外，所有顶点具有完美角分辨率，即顶点$v$处所有入射角的大小均为 $2\pi/\mathrm{deg}(v)$。我们证明了：判断给定图是否存在一种伦巴第绘制，且该绘制尊重每个顶点周围入射边的固定循环顺序，是 $\exists\mathbb{R}$-完全的。特别地，这蕴含了NP难度。尽管以往多数研究关注不同图类中伦巴第绘制的（非）存在性，我们的结果是首个关于一般图伦巴第绘制计算复杂度的结论。