Nearly Optimal Visibility Representations of Plane Graphs

Abstract

The visibility representation (VR for short) is a classical representation of plane graphs. VR has various applications and has been extensively studied in literature. One of the main focuses of the study is to minimize the size of VR. It is known that there exists a plane graph G with n vertices where any VR of G requires a size at least $(\lfloor \frac{2n}{3} \rfloor) \times (\lfloor \frac{4n}{3} \rfloor -3)$.

In this paper, we prove that every plane graph has a VR with height at most $\frac{2n}{3}+2\lceil \sqrt{n/2}\rceil$, and a VR with width at most $\frac{4n}{3}+2\lceil \sqrt{n}\rceil$. 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. However, the problem of finding VR with optimal height and optimal width simultaneously remains open.