Compact Visibility Representation of Plane Graphs

Jiun-Jie Wang (1) , Xin He (1)

1 Department of Computer Science and Engineering

The visibility representation (VR for short) is a classical representation of plane graphs. It 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 grid of size at least $\frac{2}{3}n \times (\frac{4}{3}n -3)$ (width $\times$ height). For upper bounds, it is known that every plane graph has a VR with grid size at most $\frac{2}{3}n \times (2n-5)$, and a VR with grid size at most $(n-1) \times \frac{4}{3}n$. It has been an open problem to find a VR with both height and width simultaneously bounded away from the trivial upper bounds (namely with size at most $c_h n \times c_w n$ with $c_h \lt 1$ and $c_w \lt 2$). In this paper, we provide the first VR construction with this property. We prove that every plane graph of $n$ vertices has a VR with height $\leq \HEIMX$ and width $\leq \frac{23}{12}n$. The area (height$\times$width) of our VR is larger than the area of some of previous results. However, bounding one dimension of the VR only requires finding a good $st$-orientation {\em or} a good dual $s^*t^*$-orientation of $G$. On the other hand, to bound both dimensions of VR simultaneously, one must find a good $st$-orientation {\em and} a good dual $s^*t^*$-orientation at the same time, and thus is far more challenging. Since $st$-orientation is a very useful concept in other applications, this result may be of independent interests.

Mots clés: plane graph; plane triangulation; visibility representation and st-orientation

Domaines: Complexité [cs.CC]; Algorithme et structure de données [cs.DS]

Jiun-Jie Wang, Xin He. Compact Visibility Representation of Plane Graphs. Symposium on Theoretical Aspects of Computer Science (STACS2011), Mar 2011, Dortmund, Germany. pp.141-152. ⟨hal-00573611⟩