Simple Compact Monotone Tree Drawings

Abstract

A monotone drawing of a graph G is a straight-line drawing of G such that every pair of vertices is connected by a path that is monotone with respect to some direction.Trees, as a special class of graphs, have been the focus of several papers and, recently, He and He [6] showed how to produce a monotone drawing of an arbitrary n-vertex tree that is contained in a 12n × 12n grid.All monotone tree drawing algorithms that have appeared in the literature consider rooted ordered trees and they draw them so that (i) the root of the tree is drawn at the origin of the drawing, (ii) the drawing is confined in the first quadrant, and (iii) the ordering/embedding of the tree is respected. In this paper, we provide a simple algorithm that has the exact same characteristics and, given an n-vertex rooted tree T , it outputs a monotone drawing of T that fits on a n × n grid.For unrooted ordered trees, we present an algorithms that produces monotone drawings that respect the ordering and fit in an (n + 1) × ( n 2 + 1) grid, while, for unrooted non-ordered trees we produce monotone drawings of good aspect ratio which fit on a grid of size at most 3 4 (n + 2) × 3 4 (n + 2) .

Monotone Drawings of k-Inner Planar Graphs

“…Angelini et al [1] defined the notion of slope-disjoint tree drawings and proved that a every such drawing is monotone. In order to simplify the presentation of our compact tree drawing algorithm presented in [13], we slightly extended the definition of slope-disjoint tree drawings given by Angelini et al [1]. More specifically, we called a tree drawing Γ of a rooted tree T a non-strictly slopedisjoint drawing if the following conditions hold:…”

“…We show how to extend/modify the algorithm presented in [13] so that it satisfies all properties specified in the theorem. For brevity, we refer to the algorithm of [13] as Algorithm MQuadTD (abbreviation for "Monotone Quadrant Tree Drawing"). Algorithm MQuadTD constructs a non-strictly slope-disjoint tree drawing with the root of the tree drawn at (0, 0).…”

“…Their monotone tree drawing uses a 12n × 12n grid which is asymptotically optimal as there exist trees which require at least n 9 × n 9 area [8]. In a recent paper, Oikonomou and Symvonis [13] followed a different approach from number theory and gave an algorithm based on a simple weighting method and some simple facts from geometry, that draws a tree on an n × n grid.…”

“…In Algorithm 1 we used a 2n × 2n drawing of T in the second octant in order to simplify the elongation operation. Since outerplanar graphs have no leader edges, they require no elongations and we can use instead the (first quadrant) n × n monotone tree drawing of [13], appropriately modified so that it yields a near-convex drawing.…”

Drawing a Rooted Tree as a Rooted $y-$Monotone Minimum Spanning Tree

“…see [15,Chapter 3], [4], [3]. Furthermore, drawing a rooted tree in the plane such that each pair of points is connected by a path that is monotone in some direction, in a grid of small area, is thoroughly investigated [2,9,13].…”