On the recognition and reconstruction of weighted Voronoi diagrams and bisector graphs

A weighted bisector graph is a geometric graph whose faces are bounded by edges that are portions of multiplicatively weighted bisectors of pairs of (point) sites such that each of its faces is defined by exactly one site. A prominent example of a bisector graph is the multiplicatively weighted Voronoi diagram of a finite set of points which induces a tessellation of the plane into Voronoi faces bounded by circular arcs and straight-line segments. Several algorithms for computing various types of bisector graphs are known. In this paper we reverse the problem: Given a partition $G$ of the plane into faces, find a set of points and suitable weights such that $G$ is a bisector graph of the weighted points, if a solution exists. If $G$ is a graph that is regular of degree three then we can decide in $O\left(m\right)$ time whether it is a bisector graph, where m denotes the combinatorial complexity of $G$. In the same time we can identify up to two candidate solutions such that $G$ could be their multiplicatively weighted Voronoi diagram. Additionally, we show that it is possible to recognize $G$ as a multiplicatively weighted Voronoi diagram and find all possible solutions in $O(m\log m)$ time if $G$ is given by a set of disconnected lines and circles.