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8 7 5 6 4 3 2 1 CSE 6405 Graph Drawing Text Books 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.

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Text Books

- 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.

Marks Distribution

- Attendance 10
- Participation in Class Discussions 5
- Presentation 20
- Review Report/Survey Report/
Slide Prepration 10

- Examination 55

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 PowerPoint slides for presentation.

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

Presentation Schedule

- Presentation time: 25 minutes
- Presentation will start from 5th week.

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A diagram of a computer network

Eades, Hong

Objectives of Graph Drawings

Nice drawing

structure of the graph is easy to understand

structure of the graph is difficult to understand

To obtain a nice representation of a graph so that the structure of the graph is easily understandable.

7

5

6

4

3

2

1

Objectives of Graph Drawings

Diagram of an electronic circuit

5

Wire crossings

7

4

8

3

1

2

not suitable for single layered PCB

suitable for single layered PCB

The drawing should satisfy some criterion arising from the application point of view.

Planar Drawing

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.

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

Plane graph

Straight line drawing

Plane graph

Each vertex is drawn as a point.

Each edge is drawn as a single straight line segment.

Every plane graph has a straight line drawing.

Wagner ’36 Fary ’48

Straight Line Drawing

Polynomial-time algorithm

Straight line drawing

Plane graph

Each vertex is drawn as a point.

Each edge is drawn as a single straight line segment.

Convex drawing

- 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.

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.

A 2-visibility drawing is a generalization of a visibility drawing where vertices are drawn as boxes and edges are drawn as either a horizontal line segment or a vertical line segment

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.

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.

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.

Floorplanning

VLSI Layout

Circuit Schematics

Simulating molecular structures

Data Mining

Etc…..

A

F

E

G

C

D

VLSI Floorplanning

B

B

A

A

F

F

E

E

C

G

C

G

D

D

VLSI floorplan

Interconnection graph

Dual-like graph

B

A

A

F

F

E

E

G

G

C

C

D

D

VLSI Floorplanning

B

B

A

A

F

F

E

E

C

G

C

G

D

D

VLSI floorplan

Interconnection graph

Dual-like graph

Add four corners

B

A

A

F

F

E

E

G

G

C

C

D

D

VLSI Floorplanning

B

B

A

A

F

F

E

Rectangular drawing

E

C

G

C

G

D

D

VLSI floorplan

Interconnection graph

Dual-like graph

Add four corners

corner

Rectangular drawing of

G

Plane graph G of

Output

Input

Each vertex is drawn as a point.

corner

Rectangular drawing of

G

Plane graph G of

Output

Input

Each vertex is drawn as a point.

Each edge is drawn as a horizontal or a vertical line segment.

corner

Rectangular drawing of

G

Plane graph G of

Output

Input

Each vertex is drawn as a point.

Each edge is drawn as a horizontal or a vertical line segment.

Each face is drawn as a rectangle.

Not every plane graph has a rectangular drawing.

B

B

A

A

F

F

E

Rectangular drawing

E

C

G

C

G

D

D

VLSI floorplan

Interconnection graph

B

B

A

A

F

F

E

Rectangular drawing

E

C

G

C

G

D

D

VLSI floorplan

Interconnection graph

Unwanted adjacency

Not desirable for MCM floorplanning and

for some architectural floorplanning.

B

A

F

A

F

E

G

C

E

C

G

D

D

MCM Floorplanning

Sherwani

Architectural Floorplanning

Munemoto, Katoh, Imamura

Interconnection graph

A

F

G

E

C

D

B

B

A

F

A

F

E

G

C

E

C

G

D

D

MCM Floorplanning

Architectural Floorplanning

Interconnection graph

Dual-like graph

A

F

G

E

C

D

B

B

A

F

A

F

E

G

C

E

C

G

D

D

MCM Floorplanning

Architectural Floorplanning

Interconnection graph

Dual-like graph

B

A

A

F

F

G

G

E

E

C

C

D

D

B

B

A

F

A

F

E

G

C

E

C

G

D

D

MCM Floorplanning

Architectural Floorplanning

Interconnection graph

Dual-like graph

B

A

A

F

F

G

G

E

E

C

C

D

D

Box-Rectangular drawing

B

B

A

F

A

F

E

dead space

G

C

E

C

G

D

D

MCM Floorplanning

Architectural Floorplanning

Interconnection graph

Dual-like graph

Circuit schematics

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

Planar graphs and plane graphs

different plane graphs

A plane graph is a planar graph with a fixed embedding.

An embedding is not fixed.

A planar graph may have an exponential number

of embeddings.

same planar graph

・・・・

Internet Computing

Social Sciences

Software Engineering

Information Systems

Homeland Security

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