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 segment such that the segments representing any two adjacent nodes of G are vertically visible to each other. In this 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-42/15 ⌋. 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. Moreover, if G has no degree-5 node, then our output for G is no wider than ⌈ 4n-7/3 ⌋. Also, if G is four-connected, then our output for G is no wider than n-1, matching the best known result of Kant and He. As a by-product, we obtain a much simpler proof for a corollary of Wagner’s Theorem on realizers, due to Bonichon, Saëc, and Mosbah.

Corresponding author. Address: 128 Academia Road, Section 2, Taipei 115, Taiwan. This author’s research is supported in part by NSC grant NSC-91-2213-E-001-028.