On visibility representation of plane graphs

Abstract

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 [11], Tamassia and Tollis [14] independently gave linear time VR algorithms for 2-connected plane graph. Recently, Lin et. al. reduced the width bound to [22n-42/15] [10]. In this paper, we prove that any plane graph G has a VR with width at most [13n-24/9]. For a 4-connected plane triangulation G, we give a visibility representation of G with height at most [3n/4]. In order to show that, we first show that every such graph has a canonical ordering tree with at most [n+1/2] leaves instead of the previously known bound [2n+1/3] which is of independent interest. All of them can be obtained in linear time. © Springer-Verlag 2004.