AN APPLICATION OF WELL-ORDERLY TREES IN GRAPH DRAWING

2006 ◽  
Vol 17 (05) ◽  
pp. 1129-1141 ◽  
Author(s):  
HUAMING ZHANG ◽  
XIN HE
Keyword(s):  
Graph Drawing ◽  
Linear Time ◽  
Plane Graph ◽  
Visibility Representation ◽  
Powerful Technique ◽  
Graph Generation

Well-orderly tree is a powerful technique capable of deriving new results in graph encoding, graph enumeration and graph generation [3, 5]. In this paper, by using well-orderly trees, we prove that any plane graph G with n vertices has a visibility representation with height [Formula: see text], which can be constructed in linear time. This improves the best previous bound of [Formula: see text].

scholarly journals Graph Planarity by Replacing Cliques with Paths

Algorithms ◽  
2020 ◽  
Vol 13 (8) ◽  
pp. 194
Author(s):  
Patrizio Angelini ◽  
Peter Eades ◽  
Seok-Hee Hong ◽  
Karsten Klein ◽  
Stephen Kobourov ◽  
...  
Keyword(s):  
Graph Drawing ◽  
Linear Time ◽  
Plane Graph ◽  
Topological Graph ◽  
Np Complete

This paper introduces and studies the following beyond-planarity problem, which we call h-Clique2Path Planarity. Let G be a simple topological graph whose vertices are partitioned into subsets of size at most h, each inducing a clique. h-Clique2Path Planarity asks whether it is possible to obtain a planar subgraph of G by removing edges from each clique so that the subgraph induced by each subset is a path. We investigate the complexity of this problem in relation to k-planarity. In particular, we prove that h-Clique2Path Planarity is NP-complete even when h=4 and G is a simple 3-plane graph, while it can be solved in linear time when G is a simple 1-plane graph, for any value of h. Our results contribute to the growing fields of hybrid planarity and of graph drawing beyond planarity.

A More Compact Visibility Representation

1997 ◽  
Vol 07 (03) ◽  
pp. 197-210 ◽  
Author(s):  
Goos Kant
Keyword(s):  
Planar Graph ◽  
Graph Drawing ◽  
Linear Time ◽  
Connected Components ◽  
Time And Space ◽  
Visibility Representation ◽  
Previous Time ◽  
First Time ◽  
Time Bounds ◽  
Block Tree

In this paper we present a linear time and space algorithm for constructing a visibility representation of a planar graph on an [Formula: see text] grid, thereby improving the previous bound of (2n-5)×(n-1). To this end we build in linear time the 4-block tree of a planar graph, which improves previous time bounds. Moreover, this is the first time that the technique of splitting a graph into its 4-connected components is used successfully in graph drawing

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⌋.

EVERY OUTER-1-PLANE GRAPH HAS A RIGHT ANGLE CROSSING DRAWING

2012 ◽  
Vol 22 (06) ◽  
pp. 543-557 ◽  
Author(s):  
HOOMAN REISI DEHKORDI ◽  
PETER EADES
Keyword(s):  
Empirical Evidence ◽  
Graph Drawing ◽  
Human Perception ◽  
Plane Graph ◽  
Outer Face ◽  
Input Graph ◽  
Topological Graphs ◽  
Negative Effect ◽  
Edge Crossings ◽  
Do So

There is strong empirical evidence that human perception of a graph drawing is negatively correlated with the number of edge crossings. However, recent experiments show that one can reduce the negative effect by ensuring that the edges that cross do so at large angles. These experiments have motivated a number of mathematical and algorithmic studies of “right angle crossing (RAC)” drawings of graphs, where the edges cross each other perpendicularly. In this paper we give an algorithm for constructing RAC drawings of “outer-1-plane” graphs, that is, topological graphs in which each vertex appears on the outer face, and each edge crosses at most one other edge. The drawing algorithm preserves the embedding of the input graph. This is one of the few algorithms available to construct RAC drawings.

CONVEX DRAWINGS OF PLANE GRAPHS OF MINIMUM OUTER APICES

2006 ◽  
Vol 17 (05) ◽  
pp. 1115-1127 ◽  
Author(s):  
KAZUYUKI MIURA ◽  
MACHIKO AZUMA ◽  
TAKAO NISHIZEKI
Keyword(s):  
Convex Polygon ◽  
Linear Time ◽  
Plane Graph ◽  
Decomposition Tree ◽  
Necessary And Sufficient Condition ◽  
Component Decomposition ◽  
Minimum Number ◽  
Convex Drawing ◽  
Number Of Leaves ◽  
Necessary And Sufficient

In a convex drawing of a plane graph G, every facial cycle of G is drawn as a convex polygon. A polygon for the outer facial cycle is called an outer convex polygon. A necessary and sufficient condition for a plane graph G to have a convex drawing is known. However, it has not been known how many apices of an outer convex polygon are necessary for G to have a convex drawing. In this paper, we show that the minimum number of apices of an outer convex polygon necessary for G to have a convex drawing is, in effect, equal to the number of leaves in a triconnected component decomposition tree of a new graph constructed from G, and that a convex drawing of G having the minimum number of apices can be found in linear time.

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 New Efficient Petri Nets Reductions for Parallel Programs Verification

2006 ◽  
Vol 16 (01) ◽  
pp. 101-116 ◽  
Author(s):  
Serge HADDAD ◽  
Jean-François PRADAT-PEYRE
Keyword(s):  
Petri Nets ◽  
Structural Model ◽  
Linear Time ◽  
Fundamental Property ◽  
Parallel Programs ◽  
Model Abstraction ◽  
Powerful Technique ◽  
Time Logic

Structural model abstraction is a powerful technique for reducing the complexity of a state based enumeration analysis. We present in this paper new efficient Petri nets reductions. First, we define "behavioural" reductions (i.e. based on conditions related to the language of the net) which preserve a fundamental property of a net (i.e. liveness) and any formula of the (action-based) linear time logic that does not observe reduced transitions of the net. We show how to replace these conditions by structural or algebraical ones leading to reductions that can be efficiently checked and applied whereas enlarging the application spectrum of the previous reductions. At last, we illustrate our method on a significant and typical example of a synchronisation pattern of parallel programs.

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.

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