Width-Optimal Visibility Representations of Plane Graphs

Abstract

Given an n-node plane graph G, the visibility representation of G is concerned with drawing each node of G using a horizontal line segment such that the line segments associated with any two adjacent nodes of G are vertically visible to each other. Finding most compact visibility representations of plane graphs is not only of theoretical importance but also of practical interest, and has received much attention in the community of algorithmic graph theory. In this paper, we give a linear-time algorithm to find a visibility representation of G no wider than $\lfloor\frac{4n}{3}\rfloor-2$. Our result improves upon the previously known upper bound $\frac{4n}{3}+2\lceil \sqrt{n}\rceil$, providing a positive answer to a conjecture suggested in the literature about whether an upper bound $\frac{4n}{3} + O(1)$ on the required width can be achieved for an arbitrary plane graph. In fact, our visibility representation achieves optimality in the upper bound of width because the bound differs from the previously known lower bound $\lfloor\frac{4n}{3}\rfloor-3$ only by one unit.