Re-embedding a 1-plane graph for a straight-line drawing in linear time

2021 ◽  
Author(s):  
Seok-Hee Hong ◽  
Hiroshi Nagamochi
Keyword(s):  
Linear Time ◽  
Line Drawing ◽  
Plane Graph ◽  
Straight Line

CONVEX GRID DRAWINGS OF FOUR-CONNECTED PLANE GRAPHS

2006 ◽  
Vol 17 (05) ◽  
pp. 1031-1060 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
SHIN-ICHI NAKANO ◽  
TAKAO NISHIZEKI
Keyword(s):  
Convex Polygon ◽  
Linear Time ◽  
Time Algorithm ◽  
Plane Graph ◽  
Outer Face ◽  
Plane Graphs ◽  
Line Segments ◽  
Grid Drawing ◽  
Straight Line ◽  
Grid Points

A convex grid drawing of a plane graph G is a drawing of G on the plane such that all vertices of G are put on grid points, all edges are drawn as straight-line segments without any edge-intersection, and every face boundary is a convex polygon. In this paper we give a linear-time algorithm for finding a convex grid drawing of every 4-connected plane graph G with four or more vertices on the outer face. The size of the drawing satisfies W + H ≤ n - 1, where n is the number of vertices of G, W is the width and H is the height of the grid drawing. Thus the area W · H is at most ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋. Our bounds on the sizes are optimal in a sense that there exist an infinite number of 4-connected plane graphs whose convex drawings need grids such that W + H = n - 1 and W · H = ⌈(n - 1)/2⌉ · ⌊(n - 1)/2⌋.

CONVEX DRAWINGS OF INTERNALLY TRICONNECTED PLANE GRAPHS ON O(n2) GRIDS

2010 ◽  
Vol 02 (03) ◽  
pp. 347-362 ◽  
Author(s):  
XIAO ZHOU ◽  
TAKAO NISHIZEKI
Keyword(s):  
Linear Time ◽  
Grid Point ◽  
Plane Graph ◽  
Straight Line Segment ◽  
Constant Factor ◽  
Decomposition Tree ◽  
Grid Drawing ◽  
Component Decomposition ◽  
Straight Line ◽  
Convex Drawing

In a convex grid drawing of a plane graph, every edge is drawn as a straight-line segment without any edge-intersection, every vertex is located at a grid point, and every facial cycle is drawn as a convex polygon. A plane graph G has a convex drawing if and only if G is internally triconnected. It has been known that an internally triconnected plane graph G of n vertices has a convex grid drawing on a grid of O(n3) area if the triconnected component decomposition tree of G has at most four leaves. In this paper, we improve the area bound O(n3) to O(n2), which is optimal up to a constant factor. More precisely, we show that G has a convex grid drawing on a 2n × 4n grid. We also present an algorithm to find such a drawing in linear time.

scholarly journals Convex Grid Drawings of 3-Connected Planar Graphs

1997 ◽  
Vol 07 (03) ◽  
pp. 211-223 ◽  
Author(s):  
Marek Chrobak ◽  
Goos Kant
Keyword(s):  
Planar Graphs ◽  
Linear Time ◽  
Time Algorithm ◽  
Plane Graph ◽  
Linear Time Algorithm ◽  
Convex Polygons ◽  
Line Segments ◽  
Polynomial Size ◽  
Straight Line

We consider the problem of embedding the vertices of a plane graph into a small (polynomial size) grid in the plane in such a way that the edges are straight, nonintersecting line segments and faces are convex polygons. We present a linear-time algorithm which, given an n-vertex 3-connected plane G (with n ≥ 3), finds such a straight-line convex embedding of G into a (n - 2) × (n - 2) grid.

Straight-line monotone grid drawings of series–parallel graphs

2015 ◽  
Vol 07 (02) ◽  
pp. 1550007 ◽  
Author(s):  
Md. Iqbal Hossain ◽  
Md. Saidur Rahman
Keyword(s):  
Planar Graph ◽  
Linear Time ◽  
Line Drawing ◽  
Monotone Path ◽  
Straight Line ◽  
Parallel Graph

A monotone drawing of a planar graph G is a planar straight-line drawing of G where a monotone path exists between every pair of vertices of G in some direction. Recently monotone drawings of graphs have been discovered as a new standard for visualizing graphs. In this paper we study monotone drawings of series–parallel graphs in a variable embedding setting. We show that a series–parallel graph of n vertices has a straight-line planar monotone drawing on a grid of size O(n) × O(n2) and such a drawing can be found in linear time.

scholarly journals IMPROVED ALGORITHMS FOR THE POINT-SET EMBEDDABILITY PROBLEM FOR PLANE 3-TREES

2012 ◽  
Vol 04 (01) ◽  
pp. 1250009
Author(s):  
TANAEEM M. MOOSA ◽  
M. SOHEL RAHMAN
Keyword(s):  
Efficient Algorithm ◽  
Time Complexity ◽  
Distinct Point ◽  
Line Drawing ◽  
Time Algorithm ◽  
Plane Graph ◽  
Straight Line ◽  
Point Set

In the point-set embeddability problem, we are given a plane graph G with n vertices and a point set S with the same number of points. Now the goal is to answer the question whether there exists a straight-line drawing of G such that each vertex is represented as a distinct point of S as well as to provide an embedding if one does exist. This problem has recently been solved in O(n2 log n) time for plane 3-trees. In this paper, we present a new efficient algorithm with time complexity O(n4/3+ϵ log n). We also present an O(nk4) time algorithm for the case when |S| = k > n. This is a significant improvement over the best algorithm for this case in the literature, which runs in O(nk8) time.

CANONICAL DECOMPOSITION, REALIZER, SCHNYDER LABELING AND ORDERLY SPANNING TREES OF PLANE GRAPHS

2005 ◽  
Vol 16 (01) ◽  
pp. 117-141 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
MACHIKO AZUMA ◽  
TAKAO NISHIZEKI
Keyword(s):  
Spanning Tree ◽  
Spanning Trees ◽  
Linear Time ◽  
Plane Graph ◽  
Canonical Decomposition ◽  
Sufficient Condition ◽  
Necessary And Sufficient Condition ◽  
Grid Drawing ◽  
Straight Line ◽  
Necessary And Sufficient

A canonical decomposition, a realizer, a Schnyder labeling and an orderly spanning tree of a plane graph play an important role in straight-line grid drawings, convex grid drawings, floor-plannings, graph encoding, etc. It is known that the triconnectivity is a sufficient condition for their existence, but no necessary and sufficient condition has been known. In this paper, we present a necessary and sufficient condition for their existence, and show that a canonical decomposition, a realizer, a Schnyder labeling, an orderly spanning tree, and an outer triangular convex grid drawing are notions equivalent with each other. We also show that they can be found in linear time whenever a plane graph satisfies the condition.

POLYGONAL APPROXIMATION OF DIGITAL CURVE BASED ON REVERSE ENGINEERING CONCEPT

2013 ◽  
Vol 13 (04) ◽  
pp. 1350017 ◽  
Author(s):  
KUMAR S. RAY ◽  
BIMAL KUMAR RAY
Keyword(s):  
Reverse Engineering ◽  
Linear Time ◽  
Line Drawing ◽  
Approximation Error ◽  
Zero Approximation ◽  
Polygonal Approximation ◽  
Digital Curve ◽  
Symmetric Approximation ◽  
Engineering Concept ◽  
Automatic Algorithm

This paper applies reverse engineering on the Bresenham's line drawing algorithm [J. E. Bresenham, IBM System Journal, 4, 106–111 (1965)] for polygonal approximation of digital curve. The proposed method has a number of features, namely, it is sequential and runs in linear time, produces symmetric approximation from symmetric digital curve, is an automatic algorithm and the approximating polygon has the least non-zero approximation error as compared to other algorithms.

scholarly journals Minimum Cell Connection in Line Segment Arrangements

2017 ◽  
Vol 27 (03) ◽  
pp. 159-176
Author(s):  
Helmut Alt ◽  
Sergio Cabello ◽  
Panos Giannopoulos ◽  
Christian Knauer
Keyword(s):  
Linear Time ◽  
Optimal Solution ◽  
Time Algorithm ◽  
Linear Time Algorithm ◽  
Constant Number ◽  
Line Segments ◽  
Straight Line ◽  
Minimum Number ◽  
Number Of Segments ◽  
Connection Problems

We study the complexity of the following cell connection problems in segment arrangements. Given a set of straight-line segments in the plane and two points [Formula: see text] and [Formula: see text] in different cells of the induced arrangement: [(i)] compute the minimum number of segments one needs to remove so that there is a path connecting [Formula: see text] to [Formula: see text] that does not intersect any of the remaining segments; [(ii)] compute the minimum number of segments one needs to remove so that the arrangement induced by the remaining segments has a single cell. We show that problems (i) and (ii) are NP-hard and discuss some special, tractable cases. Most notably, we provide a near-linear-time algorithm for a variant of problem (i) where the path connecting [Formula: see text] to [Formula: see text] must stay inside a given polygon [Formula: see text] with a constant number of holes, the segments are contained in [Formula: see text], and the endpoints of the segments are on the boundary of [Formula: see text]. The approach for this latter result uses homotopy of paths to group the segments into clusters with the property that either all segments in a cluster or none participate in an optimal solution.

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