A New Method for Efficiently Generating Planar Graph Visibility Representations
A planar graph visibility representation maps each vertex to a horizontal segment at a vertical position and each edge to a vertical segment at a horizontal position such that each edge segment terminates at the vertical positions of its endpoint vertices and intersects no other horizontal vertex segments. The first O(n) algorithms for producing visibility representations were presented in [4, 5]. These were based on pre-processing to compute both an st-numbering and the dual of the planar graph, which were then used with the combinatorial planar embedding to produce a visibility representation. Greater efficiency is obtained in  by eliminating the need for the planar graph dual and by re-using the pre-computed st-numbering in the PQ-tree  algorithm.