NEARLY OPTIMAL VISIBILITY REPRESENTATIONS OF PLANE GRAPHS

The visibility representation (VR for short) is a classical representation of plane graphs. The VR has various applications and has been extensively studied in the literature. One of the main focuses of the study is to minimize the size of the VR. It is known that there exists a plane graph G with n vertices, where any VR of G requires a size at least ([2n/3]) x ([4n/3]-3). For upper bounds, it is known that every plane graph has a VR with height at most [4n-1/5], and a VR with width at most [13n-24/9]. In this paper, we prove that every plane graph has a VR with height at most 2n/3 + 2[root n/2], and a VR with width at most 4n/3 + 2[root n]. These representations are nearly optimal in the sense that they differ from the lower bounds only by a lower order additive term. Both representations can be constructed in linear time. Our presentations use Schnyder's realizer to construct the st-orientations of plane graphs with special properties. As the st-orientation is a very useful concept in other applications, this result may be of independent interest.