Graph Drawing

0

Graph Drawing Tutorial

Isabel F. Cruz

Worcester Polytechnic Institute

Roberto Tamassia

Brown University

Graph Drawing

1

Introduction

Graph Drawing

2

Graph Drawing

■

models, algorithms, and systems for the

visualization of graphs and networks

■

applications to software engineering (class

hierarchies), database systems (ER-

diagrams), project management (PERT

diagrams), knowledge representation (isa

hierarchies), telecommunications (ring

covers), WWW (browsing history) ...

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Graph Drawing

3

orthogonal drawing

bend

Drawing Conventions

■

general constraints on the geometric

representation of vertices and edges

polyline drawing

planar straight-line drawing

Graph Drawing

4

strong visibility representation

planar othogonal straight-line drawing

gf

abc

e

d

g

f

a

b

d

e

c

Drawing Conventions

Graph Drawing

5

Drawing Conventions

■

directed acyclic graphs are usually drawn

in such a way that all edges “ﬂow” in the

same direction, e.g., from left to right, or

from bottom to top

■

such upward drawings effectively

visualize hierarchical relationships, such

as covering digraphs of ordered sets

■

not every planar acyclic digraph admits a

planar upward drawing

Graph Drawing

6

Resolution

■

display devices and the human eye have

ﬁnite resolution

■

examples of resolution rules:

■

integer coordinates for vertices and

bends (grid drawings)

■

prescribed minimum distance between

vertices

■

prescribed minimum distance between

vertices and nonincident edges

■

prescribed minimum angle formed by

consecutive incident edges (angular

resolution)

Graph Drawing

7

Angular Resolution

• The angular resolution

ρ

of a straight-

line drawing is the smallest angle

formed by two edges incident on the

same vertex

• High angular resolution is desirable

in visualization applications and in the

design of optical communication

networks.

•Atrivial upper bound on the angular

resolution is

where d is the maximum vertex degree.

ρ ≤

2π

d

------

Graph Drawing

8

Aesthetic Criteria

■

some drawings are better than others in

conveying information on the graph

■

aesthetic criteria attempt to

characterize readability by means of

general optimization goals

Examples

■

minimize crossings

■

minimize area

■

minimize bends (in orthogonal drawings)

■

minimize slopes (in polyline drawings)

■

maximize smallest angle

■

maximize display of symmetries

Graph Drawing

9

Trade-Offs

■

in general, one cannot simultaneously

optimize two aesthetic criteria

Complexity Issues

■

testing planarity takes linear time

■

testing upward planarity is NP-hard

■

minimizing crossings is NP-hard

■

minimizing bends in planar orthogonal

drawing:

■

NP-hard in general

■

polynomial time for a ﬁxed embedding

min # crossings max symmetries

Graph Drawing

10

Beyond Aesthetic Criteria

Graph Drawing

11

Constraints

■

some readability aspects require

knowledge about the semantics of the

speciﬁc graph (e.g., place “most

important” vertex in the middle)

■

constraints are provided as additional

input to a graph drawing algorithm

Examples

■

place a given vertex in the “middle” of

the drawing

■

place a given vertex on the external

boundary of the drawing

■

draw a subgraph with a prescribed

“shape”

■

keep a group of vertices “close” together

0

Graph Drawing Tutorial

Isabel F. Cruz

Worcester Polytechnic Institute

Roberto Tamassia

Brown University

Graph Drawing

1

Introduction

Graph Drawing

2

Graph Drawing

■

models, algorithms, and systems for the

visualization of graphs and networks

■

applications to software engineering (class

hierarchies), database systems (ER-

diagrams), project management (PERT

diagrams), knowledge representation (isa

hierarchies), telecommunications (ring

covers), WWW (browsing history) ...

1

2

3

4

5 6

7

8

9

10

11

12

13

1415

16

1718

19

20

21

22

23

24

25

26

2728

29

30

31

32

33

34

35

36

37

38

3940

41

42

43

44

45

46

47

48

49

50

51

52

53

54

55

56

57

58

59

60

61

62

63

Graph Drawing

3

orthogonal drawing

bend

Drawing Conventions

■

general constraints on the geometric

representation of vertices and edges

polyline drawing

planar straight-line drawing

Graph Drawing

4

strong visibility representation

planar othogonal straight-line drawing

gf

abc

e

d

g

f

a

b

d

e

c

Drawing Conventions

Graph Drawing

5

Drawing Conventions

■

directed acyclic graphs are usually drawn

in such a way that all edges “ﬂow” in the

same direction, e.g., from left to right, or

from bottom to top

■

such upward drawings effectively

visualize hierarchical relationships, such

as covering digraphs of ordered sets

■

not every planar acyclic digraph admits a

planar upward drawing

Graph Drawing

6

Resolution

■

display devices and the human eye have

ﬁnite resolution

■

examples of resolution rules:

■

integer coordinates for vertices and

bends (grid drawings)

■

prescribed minimum distance between

vertices

■

prescribed minimum distance between

vertices and nonincident edges

■

prescribed minimum angle formed by

consecutive incident edges (angular

resolution)

Graph Drawing

7

Angular Resolution

• The angular resolution

ρ

of a straight-

line drawing is the smallest angle

formed by two edges incident on the

same vertex

• High angular resolution is desirable

in visualization applications and in the

design of optical communication

networks.

•Atrivial upper bound on the angular

resolution is

where d is the maximum vertex degree.

ρ ≤

2π

d

------

Graph Drawing

8

Aesthetic Criteria

■

some drawings are better than others in

conveying information on the graph

■

aesthetic criteria attempt to

characterize readability by means of

general optimization goals

Examples

■

minimize crossings

■

minimize area

■

minimize bends (in orthogonal drawings)

■

minimize slopes (in polyline drawings)

■

maximize smallest angle

■

maximize display of symmetries

Graph Drawing

9

Trade-Offs

■

in general, one cannot simultaneously

optimize two aesthetic criteria

Complexity Issues

■

testing planarity takes linear time

■

testing upward planarity is NP-hard

■

minimizing crossings is NP-hard

■

minimizing bends in planar orthogonal

drawing:

■

NP-hard in general

■

polynomial time for a ﬁxed embedding

min # crossings max symmetries

Graph Drawing

10

Beyond Aesthetic Criteria

Graph Drawing

11

Constraints

■

some readability aspects require

knowledge about the semantics of the

speciﬁc graph (e.g., place “most

important” vertex in the middle)

■

constraints are provided as additional

input to a graph drawing algorithm

Examples

■

place a given vertex in the “middle” of

the drawing

■

place a given vertex on the external

boundary of the drawing

■

draw a subgraph with a prescribed

“shape”

■

keep a group of vertices “close” together

## Drawing-teaching Perspective Foundations

## Graph Drawing - General Directed

## Graph Drawing - General Undirected

## Graph Drawing - Planar Undirected

## Creating Your Own Drawing Space - Tạo không gian Vẽ riêng của bạn

## Graph Drawing - Planar Directed

## Graph Drawing - Trees

## Tài liệu Figure Drawing - Compostion doc

## Tài liệu Figure Drawing - Dynamic Figure Drawing doc

## Tài liệu Figure Drawing - Figure Anatomy pdf

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