Compact Visibility Representation of 4-Connected Plane Graphs

Abstract

The visibility representation (VR for short) is a classical representation of plane graphs. The VR has various applications and has been extensively studied. A main focus 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 $\lfloor \frac{2n}{3} \rfloor \times (\lfloor \frac{4n}{3} \rfloor -3)$. For upper bounds, it is known that every plane graph has a VR with size at most $\lfloor \frac{2}{3}n \rfloor \times (2n-5)$, and a VR with size at most $(n-1) \times \lfloor \frac{4}{3}n \rfloor$.

It has been an open problem to find a VR with both height and width simultaneously bounded away from the trivial upper bounds (namely of size c _h n ×c _w n with c _h < 1 and c _w < 2). In this paper, we provide the first VR construction for a non-trivial graph class that simultaneously bounds both the height and the width. We prove that every 4-connected plane graph has a VR with height $\leq \frac{3n}{4}+2\lceil\sqrt{n}\rceil +4$ and width $\leq \lceil \frac{3n}{2}\rceil$. Our VR algorithm is based on an st-orientation of 4-connected plane graphs with special properties. Since the st-orientation is a very useful concept in other applications, this result may be of independent interests.