The document discusses concepts and algorithms related to graph drawing, emphasizing the importance of obtaining clear visual representations of graphs, especially planar graphs. It outlines various drawing styles, objectives, properties, and applications of graph drawing such as VLSI layout and circuit schematics. Additionally, it provides details on presentation requirements for coursework in the area of graph drawing.

1. CSE 504
Graph Drawing
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8
5
7

2. Texts
• T. Nishizeki and M. S. Rahman, Planar Graph Drawing,
World Scientific, Singapore, 2004.
• G. Di Battista, P. Eades, R. Tamassia, I. G. Tollies,
Graph Drawing: Algorithms for the visualization of
Graphs, Prentice-Hall Inc., 1999.
• R. Tamassia (Ed.) Handbook of Graph Drawing and
Visualization, CRC Press, 2014.
• S. Hong et al., Beyond-Planar Graphs: Algorithmics and
Combinatorics, Report from Dagstuhl Seminar 16452,
Dagstuhl Reports, Vol. 6, Issue 11, pp. 35–62, 2017.
• S. Hong and T. Tokuyama, Beyond Planar Graphs,
Communications of NII Shonan Meetings, Springer,
2020.

3. Marks Distribution
• Attendance 5
• Paper Presentation/Book Lecture 10
• Midterm Examination 20
• Final Examination 50

4. Presentation
A paper (or a chapter of a book) from the area of
Graph Drawing will be assigned to you.
You have to read, understand and present the
paper. Use Beamer/PowerPoint slides for
presentation.

5. Presentation Format
Problem definition
Results of the paper
Contribution of the paper in respect to
previous results
Algorithm and methodology including
outline of the proofs
Future works, open problems and your
idea

6. Presentation Schedule
• Presentation time: 30 minutes
• Presentation schedule will be declared.

7. Graphs and Graph Drawings
A diagram of a computer network
ATM-SW
ATM-SW
ATM-SW
ATM-SW
ATM-SW
ATM-HUB
ATM-RT
ATM-RT
ATM-HUB
ATM-HUB
ATM-RT
ATM-RT
ATM-HUB
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
STATION
ATM-HUB
STATION
STATION

9. Objectives of Graph Drawings
To obtain a nice representation of a graph so
that the structure of the graph is easily
understandable.
structure of the graph is
difficult to understand
structure of the graph is
easy to understand
Nice drawing
Symmetric
Eades, Hong

10. The drawing should satisfy some
criterion arising from the application
point of view.
1 2 3
4
5
7
8
not suitable for single
layered PCB
suitable for single layered
PCB
1 2 3
4
6
8
5
7
Objectives of Graph Drawings
Diagram of an electronic circuit
Wire crossings

11. Drawing Styles
A drawing of a graph is planar if no two edges intersect in the
drawing.
It is preferable to find a planar drawing of a graph if the graph
has such a drawing. Unfortunately not all graphs admit planar
drawings. A graph which admits a planar drawing is called a
planar graph.
Planar Drawing

12. Polyline Drawing
A polyline drawing is a drawing of a graph in which each
edge of the graph is represented by a polygonal chain.

13. Straight Line Drawing
Plane graph

14. Straight Line Drawing
Plane graph
Straight line drawing

15. Straight Line Drawing
Each vertex is drawn as a point.
Plane graph
Straight line drawing

16. Straight Line Drawing
Each vertex is drawn as a point.
Each edge is drawn as a single straight line segment.
Plane graph
Straight line drawing

17. Straight Line Drawing
Each vertex is drawn as a point.
Each edge is drawn as a single straight line segment.
Plane graph
Straight line drawing
Every plane graph has a straight line drawing.
Wagner ’36 Fary ’48
Polynomial-time algorithm

18. Convex drawing
A straight line drawing of a plane graph G if the
boundaries of all the faces of G are convex polygon.

19. Orthogonal drawing Box-orthogonal Drawing
Rectangular Drawing Box-rectangular Drawing

20. Orthogonal drawing
An orthogonal drawing is a drawing
of a plane graph in which each edge is
drawn as a chain of horizontal and
vertical line segments

21. Box-orthogonal drawing
An box-orthogonal drawing is a drawing
of a plane graph in which each vertex is
Drawn as a rectangle box and each edge is
drawn as a chain of horizontal and
vertical line segments

22. Rectangular drawing
A rectangular drawing of a plane graph G is a
drawing of G in which each vertex is drawn as a
point, each edge is drawn as a horizontal or
vertical line segment without edge-crossings,
and each face is drawn as a rectangle

23. Box-Rectangular drawing
A box-rectangular drawing of a plane graph G is a
drawing of G on the plane such that each vertex is
drawn as a (possibly degenerate) rectangle, called a
box, and the contour of each face is drawn as a
rectangle

24. Grid drawing
A drawing of a graph in which vertices and bends are
located at grid points of an integer grid

25. Grid Drawing

26. • When the embedding has to be drawn on a raster
device, real vertex coordinates have to be mapped to
integer grid points, and there is no guarantee that a
correct embedding will be obtained after rounding.
• Many vertices may be concentrated in a small region
of the drawing. Thus the embedding may be messy,
and line intersections may not be detected.
• One cannot compare area requirement for two or more
different drawings using real number arithmetic, since
any drawing can be fitted in any small area using
magnification.
Grid Drawing

27. Visibility drawing
A visibility drawing of a plane graph G is a drawing of G where
each vertex is drawn as a horizontal line segment and each
edge is drawn as a vertical line segment.
The vertical line segment representing an edge must connect
points on the horizontal line segments representing the end
vertices.

28. Properties of graph drawing
Area. A drawing is useless if it is unreadable. If the used area
of the drawing is large, then we have to use many pages, or we
must decrease resolution, so either way the drawing becomes
unreadable. Therefore one major objective is to ensure a small
area. Small drawing area is also preferable in application
domains like VLSI floorplanning.
Aspect Ratio. Aspect ratiois defined as the ratio of the length of
the longest side to the length of the shortest side of the smallest
rectangle which encloses the drawing.

29. Bends. At a bend, the polyline drawing of an edge
changes direction, and hence a bend on an edge
increases the difficulties of following the course
of the edge. For this reason, both the total number of
bends and the number of bends per edge should be kept
small.
Crossings. Every crossing of edges bears the potential
of confusion, and therefore the number of crossings
should be kept small.
Shape of Faces. If every face has a regular shape in a
drawing, the drawing looks nice. For VLSI floorplanning,
it is desirable that each face is drawn as a rectangle.

30. Symmetry. Symmetry is an important aesthetic criteria in
graph drawing. A symmetryof a two-dimensional figure is
an isometry of the plane that fixes the figure.
Angular Resolution. Angular resolution is measured by
the smallest angle between adjacent edges in a drawing.
Higher angular resolution is desirable for displaying a
drawing on a raster device.

31. Applications of Graph Drawing
Floorplanning
VLSI Layout
Circuit Schematics
Simulating molecular structures
Data Mining
Etc…..

32. VLSI Layout

33. E
A
B
C
F
G
D
VLSI Floorplanning
Interconnection graph

34. E
A
B
C
F
G
D
A
B
E
C
F
G
D
VLSI Floorplanning
Interconnection graph VLSI floorplan

35. E
A
B
C
F
G
D
A
B
E
C
F
G
D
VLSI Floorplanning
Interconnection graph VLSI floorplan

36. E
A
B
C
F
G
D
A
B
E
C
F
G
D
VLSI Floorplanning
Interconnection graph VLSI floorplan

37. E
A
B
C
F
G
D
A
B
E
C
F
G
D
VLSI Floorplanning
A
B
E
C
F
G
D
Interconnection graph VLSI floorplan
Dual-like graph

38. E
A
B
C
F
G
D
A
B
E
C
F
G
D
VLSI Floorplanning
A
B
E
C
F
G
D
A
B
E
C
F
G
D
Interconnection graph VLSI floorplan
Dual-like graph Add four corners

39. E
A
B
C
F
G
D
A
B
E
C
F
G
D
VLSI Floorplanning
A
B
E
C
F
G
D
A
B
E
C
F
G
D
Interconnection graph VLSI floorplan
Dual-like graph Add four corners
Rectangular
drawing

40. Rectangular Drawings
Plane graph G of 3


Input

41. Rectangular Drawings
Rectangular drawing of G
Plane graph G of 3


Input Output
corner

42. Rectangular Drawings
Rectangular drawing of G
Plane graph G of 3


Each vertex is drawn as a point.
Input Output
corner

43. Rectangular Drawings
Rectangular drawing of G
Plane graph G of 3


Each edge is drawn as a horizontal or a vertical
line segment.
Each vertex is drawn as a point.
Input Output
corner

44. Rectangular Drawings
Rectangular drawing of G
Plane graph G of 3


Each edge is drawn as a horizontal or a vertical
line segment.
Each face is drawn as a rectangle.
Each vertex is drawn as a point.
Input Output
corner

45. Not every plane graph has a rectangular drawing.

46. E
A
B
C
F
G
D
A
B
E
C
F
G
D
VLSI Floorplanning
Interconnection graph VLSI floorplan
Rectangular
drawing

47. E
A
B
C
F
G
D
A
B
E
C
F
G
D
VLSI Floorplanning
Interconnection graph VLSI floorplan
Rectangular
drawing
Unwanted adjacency
Not desirable for MCM floorplanning and
for some architectural floorplanning.

48. A
B
E C
F
G
D
MCM Floorplanning
Sherwani
Architectural Floorplanning
Munemoto, Katoh, Imamura
E
A
B
C
F
G
D
Interconnection graph

49. A
B
E C
F
G
D
MCM Floorplanning
Architectural Floorplanning
E
A
B
C
F
G
D
Interconnection graph

50. MCM Floorplanning
Architectural Floorplanning
A
E
B
F
G
C
D
E
A
B
C
F
G
D
Interconnection graph
Dual-like graph
A
B
E C
F
G
D

51. MCM Floorplanning
Architectural Floorplanning
A
E
B
F
G
C
D
E
A
B
C
F
G
D
Interconnection graph
Dual-like graph
A
B
E C
F
G
D

52. MCM Floorplanning
Architectural Floorplanning
A
E
B
F
G
C
D
E
A
B
C
F
G
D
A
E
B
F
G
C
D
Interconnection graph
Dual-like graph
A
B
E C
F
G
D

53. MCM Floorplanning
Architectural Floorplanning
A
E
B
F
G
C
D
E
A
B
C
F
G
D
A
E
B
F
G
C
D
Box-Rectangular
drawing
Interconnection graph
Dual-like graph
A
B
E C
F
G
D
dead space

54. Applications
Entity-relationship diagrams
Flow diagrams

55. Applications
Circuit schematics
Minimization of bends reduces the number of “vias” or
“throughholes,” and hence reduces VLSI fabrication costs.

56. A planar graph
planar graph non-planar graph

57. Planar graphs and plane graphs
An embedding is not fixed.
A planar graph may have an exponential number
of embeddings.
A plane graph is a planar graph with a fixed embedding.
different plane graphs
same planar graph
・・・・

58. Good Route Planning
Monotone along
the direction
59

59. ◮ Each vertex is drawn at a grid point on an integer grid.
◮ Each edge is drawn as a straight-line segment without edge
crossings.
◮ At least a monotone path exists between every pair of
vertices.
Definition of monotone drawing
60

60. ◮ Each vertex is drawn at a grid point on an integer grid.
◮ Each edge is drawn as a straight-line segment without edge
crossings.
◮ At least a monotone path exists between every pair of
vertices.
Definition of monotone drawing
61

61. ◮ Each vertex is drawn at a grid point on an integer grid.
◮ Each edge is drawn as a straight-line segment without edge
crossings.
◮ At least a monotone path exists between every pair of
vertices.
Definition of monotone drawing
62

62. •◮ Each vertex is drawn at a grid point on an integer grid.
◮ Each edge is drawn as a straight-line segment without edge
crossings.
◮ At least a monotone path exists between every pair of
vertices.
•Definition of monotone drawing
63

63. ◮ Each vertex is drawn at a grid point on an integer grid.
◮ Each edge is drawn as a straight-line segment without edge
crossings.
◮ At least a monotone path exists between every pair of
vertices.
Definition of monotone drawing
64

64. ◮ Each vertex is drawn at a grid point on an integer grid.
◮ Each edge is drawn as a straight-line segment without edge
crossings.
◮ At least a monotone path exists between every pair of
vertices.
Definition of monotone drawing
65

65. Road network in mega city planning Robot movement with obstacles
Application of monotone drawing
66