Improved compact visibility representation of planar graph via Schnyder's realizer

Abstract

Let G be an n-node planar graph. In a visibility representation of G, each node of G is represented by a horizontal line segment such that the line segments representing any two adjacent nodes of G are vertically visible to each other. In the present paper we give the best known compact visibility representation of G. Given a canonical ordering of the triangulated G, our algorithm draws the graph incrementally in a greedy manner. We show that one of three canonical orderings obtained from Schnyder's realizer for the triangulated G yields a visibility representation of G no wider than [22n-40/15]. Our easy-to-implement O(n)-time algorithm bypasses the complicated subroutines for four-connected components and four-block trees required by the best previously known algorithm of Kant. Our result provides a negative answer to Kant's open question about whether [3n-6/2] is a worst-case lower bound on the required width. Also, if G has no degree-three ( respectively, degree-five) internal node, then our visibility representation for G is no wider than [4n-9/3]( respectively, [4n-7/3]). Moreover, if G is four-connected, then our visibility representation for G is no wider than n - 1, matching the best known result of Kant and He. As a by-product, we give a much simpler proof for a corollary of Wagner's theorem on realizers due to Bonichon, Le Saec, and Mosbah.