New theoretical bounds of visibility representation of plane graphs

In a visibility representation (VR for short) of a plane graph G, each vertex of G is represented by a horizontal line segment such that the line segments representing any two adjacent vertices of G are joined by a vertical line segment. Rosenstiehl and Tarjan, Tamassia and Tollis independently gave linear time VR algorithms for 2-connected plane graph. Afterwards, one of the main concerns for VR is the size of VR. In this paper, we prove that any plane graph G has a VR with height bounded by ⌊5n/6⌋. This improves the previously known bound ⌈15n/16⌉. We also construct a plane graph G with n vertices where any VR of G require a size of (⌊2n/3⌋) × (⌊4n/3⌋ - 3). Our result provides an answer to Kant's open question about whether there exists a plane graph G such that all of its VR require width greater that cn, where c > 1.