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, Saëc, and Mosbah.
Given a graph G = (V, E) and a set T of non-negative integers containing 0, a T -coloring of G is an integer function f of the vertices of G such that |f(u) − f(v)| / ∈ T whenever uv ∈ E. The edge-span of a Tcoloring f is the maximum value of |f(u) − f(v)| over all edges uv, and the T -edge-span of a graph G is the minimum value of the edge-span among all possible T -colorings of G. This paper discusses the T -edge span of the folded hypercube network of dimension n for the k-multiple-of-
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, Saëc, and Mosbah.
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