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：在图的隆巴迪绘制中，顶点被绘制为点，边被绘制为连接各自端点的圆弧。此外，所有顶点均具有完美的角分辨率，即入射到顶点 $v$ 的所有角的大小均为 $2\pi/\mathrm{deg}(v)$。我们证明，判断给定图是否允许一种隆巴迪绘制（该绘制尊重每个顶点处入射边的固定循环顺序）是 $\exists\mathbb{R}$-完备的。特别地，这意味着该问题具有NP难度。虽然以往大多数工作研究了不同图类中隆巴迪绘制的（不）存在性，我们的结果是关于一般图隆巴迪绘制计算复杂性的首个成果。