# Building vertex-edge visibility graph among polygonal obstacles on 2d plane

I want to implement algorithm for computing vertex-edge visibility graph among polygonal obstacles, but I can't find any description or scientific paper describing such algorithm. Currently I understand how to extend Lee's algorithm for vertex-vertex visibility graph to compute vertex-edge graph, but I want to achieve better than $O(n^2 log(n))$ complexity. I read paper of Mount and Ghosh about their $O(nlog(n))$ algorithm based on triangulation of free space, and seems it allows extension to produce vertex-edge graph. But that algo has too complicated implementation :)

I wonder if there is a way to extend Overmars and Welzl algorithm $O(n^2)$ or algorithm with similar complexity to produce vertex-edge visibility graph? It's implementation seems to be quite simple. Point me on some papers, links, or describe it here if you can.

I'm not sure if this is what you're after, but here's an $O(n + k)$ approach from J. O'Rourke and I. Streinu, Computational Geometry 10 (1998), pp. 105-120:
$G_{VE}$ may be constructed in $O(n + k)$ time for a polygon with $n$ vertices and $k$ visibility edges, by a slight modification of Hershberger's algorithm that constructs $G_v$ in this time bound [8] from a polygon triangulation. Supplementing with Chazelle's linear-time triangulation algorithm achieves the claimed bound.