Any system of bisectors (in the sense of abstract Voronoi diagrams) defines an arrangement of simple curves in the plane. We define Voronoi-like graphs on such an arrangement, which are graphs whose vertices are locally Voronoi. A vertex $v$ is called locally Voronoi, if $v$ and its incident edges appear in the Voronoi diagram of three sites. In a so-called admissible bisector system, where Voronoi regions are connected and cover the plane, we prove that any Voronoi-like graph is indeed an abstract Voronoi diagram. The result can be seen as an abstract dual version of Delaunay's theorem on (locally) empty circles. Further, we define Voronoi-like cycles in an admissible bisector system, and show that the Voronoi-like graph induced by such a cycle $C$ is a unique tree (or a forest, if $C$ is unbounded). In the special case where $C$ is the boundary of an abstract Voronoi region, the induced Voronoi-like graph can be computed in expected linear time following the technique of [Junginger and Papadopoulou SOCG'18]. Otherwise, within the same time, the algorithm constructs the Voronoi-like graph of a cycle $C'$ on the same set (or subset) of sites, which may equal $C$ or be enclosed by $C$. Overall, the technique computes abstract Voronoi (or Voronoi-like) trees and forests in linear expected time, given the order of their leaves along a Voronoi-like cycle. We show a direct application in updating a constraint Delaunay triangulation in linear expected time, after the insertion of a new segment constraint, simplifying upon the result of [Shewchuk and Brown CGTA 2015].

翻译：任意平分线系统（在抽象Voronoi图的意义下）定义了平面上的简单曲线排列。我们在此类排列上定义Voronoi类图，即顶点具有局部Voronoi性质的图。若顶点$v$及其关联边出现在三个站点的Voronoi图中，则称$v$为局部Voronoi顶点。在所谓可容许平分线系统中（其中Voronoi区域连通且覆盖整个平面），我们证明任意Voronoi类图本质上即为抽象Voronoi图。该结果可视为Delaunay（局部）空圆定理的抽象对偶形式。进一步地，我们在可容许平分线系统中定义Voronoi类环，并证明由此类环$C$诱导的Voronoi类图是唯一的树（若$C$无界则为森林）。当$C$为抽象Voronoi区域的边界时，可通过[Junginger and Papadopoulou SOCG'18]的技术在期望线性时间内计算诱导Voronoi类图。否则，算法在相同时间内为同一站点集（或其子集）构造环$C'$的Voronoi类图，其中$C'$可能等于$C$或被$C$包围。总体而言，该技术在给定叶子沿Voronoi类环排列顺序的条件下，以期望线性时间计算抽象Voronoi（或Voronoi类）树与森林。我们展示了该技术的一个直接应用：在插入新线段约束后，以期望线性时间更新约束Delaunay三角剖分，该结果简化了[Shewchuk and Brown CGTA 2015]的结论。