layouts for graph drawing contests 1995-2001

UNIVERSITY OF LJUBLJANA

INSTITUTE OF MATHEMATICS, PHYSICS AND MECHANICS

DEPARTMENT OF THEORETICAL COMPUTER SCIENCE

JADRANSKA 19, 1 000 LJUBLJANA, SLOVENIA

Preprint series, Vol. 40 (2002), 568/2

LAYOUTS FORGRAPH DRAWING CONTESTS

1995-2001

Vladimir Batagelj, Andrej Mrvar

ISSN 1318-4865

Version 1: May 13, 1998Version 2: December 27, 2001

Math.Subj.Class.(2000): 05 C 90, 68 R 10, 76 M 27, 68 U 05,05 C 50, 05 C 85, 90 C 27, 92 H 30, 92 G 30, 93 A 15, 62 H 30.

Supported by the Ministry of Education, Science and Sport of Slovenia,Project J1-8532.

Address: Vladimir Batagelj, University of Ljubljana, FMF, Departmentof Mathematics, and IMFM Ljubljana, Department of TCS, Jadranskaulica 19, 1 000 Ljubljana, Slovenia

e-mail: [email protected]

Ljubljana, December 27, 2001

Layouts forGraph Drawing Contests

1995-2001

Vladimir BatageljDepartment of Mathematics, University of Ljubljana

[email protected]

Andrej MrvarFaculty of Social Sciences, University of Ljubljana

[email protected]

Graph drawing contests

Since 1994 graph drawing contests are organized as a part of the Graph Drawing Confer-ences. The rules and data are described on Internet and participants send their drawingstill the specified date. They can use any technique to get layouts of graphs. The primaryjudging criterion is how well the drawings convey the information in the graphs: vertexidentifiers, vertex types, and vertex interconnectivity. A secondary criterion is the degreeto which manual editing was employed to produce the layout: the less manual intervention,the better.

The purpose of the contests is to monitor and challenge the current state of the art ingraph-drawing technology.

Data (vertices and lines) for 3 or 4 graphs are given each year. A winning entry for eachgraph is chosen by a panel of experts.

In this paper we collected our submissions to the contests in the years 1995–2001. Theyare available also at

http://vlado.fmf.uni-lj.si/pub/gd/gd95.htmand the original data (and their versions in Pajek format) at

http://vlado.fmf.uni-lj.si/pub/networks/data/gd/gd.htmThere you will find also a longer version of this collection (some pictures are to large forthe format of the preprint publication) and pictures in some (dynamical) graphical formats(VRML, SVG) that can not be adequately reproduced in the paper form.

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Layouts for Graph Drawing Contest 1995

In 1995 the Graph Drawing Conference was helt in Passau and the contest was organizedby Peter Eades and Joe Marks. Rules and data are described at:

http://www.uni-passau.de/agenda/gd95/contest.html

Graph A95� First layout was obtained automatically using our program COORD (positioning ver-tices on the rectangular net so that the number of crossings of lines is as low aspossible).

� Manual editing was performed to reposition vertices using our graph picture editorDRAW.

Graph B95� Analysing graph B using our program RELCALC two central vertices were found (1and 34).

� Feasible possitions for vertices were generated (two families of concentric circles).

� Vertices 1 and 34 were fixed in the centre of the concentric circles mentioned.

� Other vertices were automatically positioned around using program COORD so thatthe total length of the lines was minimized.

� Layout was then edited manually using DRAW to reposition vertices to minimizecrossings (concentric circles cannot be seen any more).

The layout was awarded the honorable mention.

Graph C95� First layout was obtained automatically using our program ENERG (minimisation ofENERGy).

� Some manual editing using DRAW was performed to reposition vertices.

The layout was awarded the first prize.

See: http://vlado.fmf.uni-lj.si/pub/gd/gd95.htmThe complete report of the contest is available in [6] and:

http://www.merl.com/reports/TR95-14/index.html

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Figure 1: Graph A95.

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Figure 2: Graph B95 (honorable mention).

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Figure 3: Graph C95 (first prize).

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In 1996 Graph Drawing Conference was helt in Berkeley, and the contest was organized byPeter Eades, Joe Marks, and Stephen North. Rules and data are desribed at:

http://portal.research.bell-labs.com/orgs/ssr/people/north/contest.html

Graph B96� First layout was obtained automatically using our program COORD. (positioning ver-tices on the rectangular net so that the number of crossings of lines is as low aspossible).

� Manual editing was performed to reposition vertices using our graph picture editorDRAW.

The layout was awarded the honorable mention.

Graph C96� Analysing graph C using our program RELCALC two parts and 2 connecting verticeswere found.

� Each part was handled separately using our program ENERG (minimisation of en-ergy). One part consists of a lattice structure, and the second of a planar graph ofcylindric symmetries. This spatial picture was realized in VRML (produced from thedescription in our graph description language NetML based on SGML).� Some manual editing was done for planar representation.


Graph D96� Analysing graph D using program RELCALC 17 important vertices of ’kernel graph’were found.

� The first layout for these 17 vertices was obtained automatically using program COORD.� Other vertices were added to the obtained picture.

� Some manual editing using DRAWwas performed to reposition vertices to avoid cross-ings.

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Figure 4: Graph B96 (honorable mention).

See: http://vlado.fmf.uni-lj.si/pub/gd/gd96.htmThe complete report of the contest is available in [10] and:

http://www.merl.com/reports/TR96-24/index.html

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Figure 6: Graph C96 / VRML snapshot.

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Figure 7: Graph C96 with cylinder / VRML snapshot.

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Figure 8: Graph D96.

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In 1997 Graph Drawing Conference was helt in Rome and the contest was organised byPeter Eades, Joe Marks, and Stephen North. Rules and data are described at:http://portal.research.bell-labs.com/orgs/ssr/people/

north/contest.html

Graph A97� First layout was obtained automatically using draw/eigenvalues option in pro-gram Pajek. Manual editing was used to reposition vertices in the grid to obtainorthogonal layout in plane.


� Manual editing in 3D was performed to get orthogonal embeddings in space: mini-mal, symmetric and cube.

Graph B97� Analysing graph B using our program MODEL we obtained (almost) regular partitionin 3 classes. The third class contains only vertex Harmony Central. The secondclass, represented by squares, contains 11 vertices that are connected only to thevertices in the class 1 (represented by circles). Vertices in class 1 are also connectedto other vertices in the same class. We first drew all vertices in the class 1 in the centerand vertices in class 2 separately – using class shrinking and circular drawing optionsin Pajek. Afterward we manually moved vertices of class 1 connected to only onevertex of class 2 close to this vertex. Finally we manually arranged the remainingvertices of class 1.

� We transformed given similarities � on arcs to dissimilarities�� and applied

Ward’s hierarchical clustering method to the obtained dissimilarity matrix. We pro-duced a clustering into 12 clusters, shrank the graph using Pajek, and draw the ob-tained skeleton minimizing the number of crossings. Finally we manually arrangedthe vertices of original graph.


See alsohttp://vlado.fmf.uni-lj.si/pub/gd/gd97.htm

The complete report is available in [7] or athttp://www.merl.com/reports/TR97-16/index.html

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Figure 9: Graph A97 – orthogonal layout in plane (first prize).

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Figure 10: Graph A97 – 3D minimal / VRML snapshot.

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Figure 11: Graph A97 – 3D symmetric / VRML snapshot.

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Figure 12: Graph A97 – 3D cube / VRML snapshot.

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1997 North American Gs Develop

Edirol Online

Virtual Sound Canvas Online

Microsoft Corporation Home Pag

Microsoft Netshow Events Calen

Wavelet Netcare

Wavelab

Wavelet Resources

IBM Corporation

Kasparov Vs. DeepBlue: The Rem

Guest Essay: David Stork

Boston At Night: Feature Story

Intranet Executive: Digital Id

Netscape Navigator Family

Netscape Devedge: Library

Netscape Devedge: Library 1

Netscape Devedge: Courses & Co


Four Corners Hantavirus

Bunyaviridae

INI International: Grocery Sec

INI International: Vitamins

INI International: Candy Secti

Journal EntryCelery

Onions

The Biting Truth

Musky Have Teeth

404 Not Found

Cold Front Tactics

Identifying Vegetation

Tackle Box Survival - Surface

Sandell & Associates Surface L

Musky Muskie Lure Components

Young Designer Of The Month

The London College Of Fashion

Designercity

Designercity1

Designercity2

Designercity3

Designercity4

Designercity5

Designercity6

Nada138-954

Janet’s Horsin’ Around On The

Equinet(tm) - Comments

Harmony Central: Midi Tools

Figure 13: Graph B97 – regular equivalence.

18

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Figure 14: Graph B97 – Dendrogram for Ward’s method.

19

1997 North American Gs Develop

Edirol Online

Virtual Sound Canvas

Microsoft Corporation Home Pag

Microsoft Netshow Events Calen

Wavelet NetcareWavelab

Wavelet Resources

IBM Corporation

Kasparov Vs. DeepBlueGuest Essay: D. Stork

Boston At Night: Feature StoryIntranet Executive: Digital Id

Netscape Navigator Family

Netscape Devedge: Library

Netscape Devedge: Library 1Netscape Devedge: Courses & Co


Four Corners Hantavirus

Bunyaviridae

INI International: Grocery Sec

INI International: VitaminsINI International: Candy Secti

Journal EntryCeleryOnions

The Biting Truth

Musky Have Teeth

404 Not Found

Cold Front Tactics

Identifying Vegetation

Tackle Box Survival - Surface Sandell & Associates Surface L

Musky Muskie Lure Components

Young Designer Of The Month

The London College Of Fashion

Designercity

Designercity1Designercity2



Nada138-954

Janet’s Horsin’ Around On The

Equinet(tm) - Comments

Harmony Central: Midi Tools

Figure 15: Graph B97 – Ward’s clustering (first prize).

20


In 1998 Graph Drawing Conference was helt in Montreal and the contest was organised byPeter Eades, Joe Marks, Petra Mutzel, and Stephen North. Rules and data are described at:

http://gd98.cs.mcgill.ca/contest/All work was done using our program package Pajek, which is freely available at:

http://vlado.fmf.uni-lj.si/pub/networks/pajek/

Graph A98

The data for graph A consist of addition and deletion operations that specify how the graphchanges over time. In our solution addition and deletion operations were performed usingmacro language which is recognized by program package Pajek. Then we used Mi-crosoft Camcorder (free software) to record the process. Selected layouts in threedifferent time points are shown.

Graph B98

Symmetries in graph (with some exceptions) can easily be noticed. We started the draw-ing using the layout obtained using Fruchterman-Reingold spring embedder. Later we usedmanual editing to maximize symmetries and made some displacements according to differ-ent sizes and shapes of nodes. Pajek was used to do all the work.


Graph C98

Using spring embedders we could not get any nice layouts. Analysing graph we found thatit is a symmetric cubic graph. Later we used eigenvectors approach and computed eigenvec-tors of neighbourhood matrix that correspond to the largest eigenvalue (which is multiple).In this way we got nice symmetric picture in space (3D). According to symmetries (equiva-lences) some of the nodes are drawn on the same positions and some are overlapping whenselecting the certain view to see the symmetries. Since some vertices are overlapping webuilt a list of overlapping vertices drawn in different colors:

Graph D98

There were no data available for this graph. The participants could send any picture thatis inspired or related to graph drawing. We decided to generate graph from dictionary andsent some interesting subgraph of it.

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Table 1: Overlapping vertices in graph C98

Group Color Vertices1. Yellow 13, 762. Green 9, 12, 33,1063. Pink 39, 65, 88,1124. Blue 5, 20, 51, 635. Fuchsia 68, 75, 85, 926. White 36, 997. Orange 21, 428. Purple 23, 24, 55, 899. NavyBlue 30, 44, 54, 57

10. TealBlue 3, 11, 43,10111. OliveGreen 1, 18, 64, 74, 91,103,104,10712. Gray 19, 26, 47, 5913. Black 17, 40, 70, 7914. Maroon 48, 5315. LightGreen 25, 32, 35, 60, 66, 71, 73,10516. Cyan 10,10217. Yellow 2, 22, 81, 9018. Green 14, 58, 61, 8319. Pink 4, 15, 46, 52, 77, 78,109,11020. Blue 7, 8, 27, 4121. Fuchsia 6, 50, 67, 9622. White 82, 84, 95, 9723. Orange 31, 8024. Purple 16, 29, 98,10825. NavyBlue 38, 72, 86, 8726. TealBlue 37, 4527. OliveGreen 28, 62, 93,11128. Gray 34, 49, 56,10029. Black 69, 94

Large graph can be generated from words in a dictionary. We constructed a graph inwhich two words are connected iff one can be obtained from the other by changing sin-gle character (e. g. WORK – WORD) or by adding/deleting one character (e.g. EVER– FEVER). Then we took english words graph-drawing-contest, find all shortestpaths between graph and drawing and between drawing and contest, draw the ob-tained graph using layers option in Pajek (layers are determined by distance from draw-ing). Additionally we added two puzzles:� one difficult: find shortest paths alone,

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Figure 16: Graph C98.

� one easy: find only missing words on the paths.

See also: http://vlado.fmf.uni-lj.si/pub/gd/gd98.htm

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In 1999 Graph Drawing Conference was helt in Stirin (Czeck Republic) and the contestwas organised by Franz Brandenburg, Michael Juenger, Joe Marks, Petra Mutzel, and FalkSchreiber. Rules and data are described at:

http://www.ms.mff.cuni.cz/acad/kam/conferences/GD99/contest/rules.html

All work was done using our program package Pajek, which is freely available at:http://vlado.fmf.uni-lj.si/pub/networks/pajek/

Graph A99

Graph data were transformed to Pajek format, which enables to handle graph changes overtime. The following coding of actors is used:

� Shapes

– circle: =f= female

– triangle: =m= male

– yellow box: =b= female+male (Ernst+Lisl Wiesenhuber)

– green box: =a= women+grandchildren (Julia von der Marwitz)

– black diamond: =w= widowed

– pink diamond: =d= divorced marriage

– orange diamond: =m= still existing marriage

– empty: =i= invisible node (white)

� Sizes

– 0.8 - big

– 0.4 - small

– 0.6 - artificial

� Colors

– =y= unactive characters (yellow)

– =a= 2 persons in 2 different colors (green)

– =b= active characters (blue)

– =d= divorced marriage (pink)

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– =w= characters that never showed up personally (white)

– =m= still existing marriage (orange)

– =g= already dead characters (grey)

– =w= widowed (black)� Border widths

– =p= picture - border width = 2

– =s= symbol - border width = 0.5

Drawing of the current situation We started the drawing using the layout obtainedby Kamada-Kawai spring embedder. Later we used manual editing to arrange them onrectangular net.

Development of the graph over time We used the Pajek option for drawing graphsin different time points. Only time points where at least one vertex or one line changesaccording to last layout were drawn (e.g. 1, 2, 4, 5,...). After each change time the layout ofthe new graph was optimized starting with the previous positions.


Graph B99

We used eigenvectors approach and computed eigenvectors of Laplacean matrix that corre-spond to the 1st, 2nd and 5th largest eigenvalue. No additional manual editing was used onthe obtained spatial picture. Suitable view was selected to get planar EPS picture. Picturewas exported to formats for the following 3D viewers: VRML (CosmoPlayer), kinemages(Mage), and MDLMOLfile (Chime).

The layout was awarded the second prize.

Graph C99

We started the drawing using the layout obtained by Fruchterman-Reingold spring embed-der. Later we used manual editing to maximize ’rectangularity’ and made some displace-ments to accommodate for different sizes of nodes.

See: http://vlado.fmf.uni-lj.si/pub/gd/gd99.htm .The complete report of the contest is available in [8].

31

Rosi Koch

Zorro Franz Josef Pichelsteiner

Onkel Franz Franz Wittich

Gung Pahm Kien

Urzula Wienicki

Gabi Zenker-geb.Skawowski

Phil Seegers

Hans Wilhelm Huelsch

Hajo Scholz

Olli Klatt

Erich-Schiller

Helga Beimer Schiller

Hans Beimer Anna Ziegler

Andy Zenker

Berta Griese

Lisa Hoffmeister

Marion Beimer

Klaus Beimer

Dr. Eva-Maria Sperling

Boris Ecker

Valerie Ecker

Iffi Zenker

Momo Sperling

Dr. Ahmet Dagdelen

Else Kling

Paolo Varese

Dani Schmitz

Philipp Sperling

Tanja Schildknecht-Dressler

Dr. Ludwig Dressler

Mary KlingOlaf Kling

Fausto Rossini

Frank DresslerElena Sarikakis

Isolde Pavarotti

Kaethe Georg Eschweiler

Carsten Floeter Theo Klages

Beate Sarikakis

Vasily Sarikakis

Rolf Sattler Bolle Guenther Bollmann

Hilmar Eggers

Wilhelm Loesch

Marlene Schmitt

Friedemann Traube

Wanda Winicki

Pat Wolffson

Winifred Snyder

Irina Winicki

Sophie ZieglerTom Ziegler

Sarah Ziegler

Paula Madalena Francesca Winicki

Canan Dagdelen

Herr Panowski

Giovanna Varese

Marcella Varese

Nico Zenker

Gina Varese

Chromo Hoyonda

Alfredo

Francesco

Professor Dr. Rudolf Tenge-Wegemmann

Frau Horowitz

Harry

Zeki Kurtalan

Fritjof Lothar Boedefeld

Gundel Koch

Dr. Otto Pichelsteiner

Leo Klamm

Rita Bassermann

Simone Fitz

Dora Wittich

Martha

Bruno Skabowski

Sue

Theresa Zenker

Inge Kling

Ernst+Lisl Wiesenhuber

gesch.Skabowski*19.10.1927

*09.08.1963

*24.07.1913

*23.10.1958

*10.02.1963

verw.Zimmermann*23.05.1960

*14.04.1961

*29.02.1952

*09.06.1944

*06.01.1976

*23.07.1944

geb.Wittich*24.03.1940

*03.10.1943 geb.Jenner*14.07.1959

*03.07.1947

geb.Nolte*25.06.1941

*19.10.1981

*21.05.1969

*17.10.1078

*06.08.1947

*17.06.1965

geb.Zenker*08.10.1975

*19.08.1978

*01.04.1975

*14.05.1922

*14.09.1977

geb.Schildknecht*19.06.1970

*27.06.1933

geb.Dankor*30.01.1969*10.11.1955

*31.10.1966*08.11.1939

verw.Panowak*27.11.1936

*05.03.1966

geb.Floeter*17.08.1970

*14.02.1963

*02.07.1942

geb.Kollin*17.07.1938

*05.05.1972

*18.10.1990

*25.07.1991*27.07.1989

*22.10.1987

*29.08.1996

*12.05.1994*25.10.1970

*06.04.1940

*22.05.1962

+1964

Unbekannter_Russe

+1986

*1947+1964

*1955+1965

*19.09.1996

*10.05.1968+29.08.1991

*30.01.1992

*14.11.1991

*1969+1986

*19.04.1996

*13.03.1994

*18.07.1996

*24.07.1997

*26.11.1987+18.08.1996Figure

23:G

raphA

99,currentsituation.

32

Lydia Nolte

Gung Pahm Kien



Hans Beimer

Berta Griese

Marion Beimer

Benny BeimerKlaus Beimer

Franz Schildknecht

Henny Schildknecht

Stefan Nossek

Egon Kling

Else Kling

Tanja Schildknecht-DresslerDr. Ludwig Dressler

Vasily Sarikakis

Chris Barnsteg

Elfie Kronmayr

Sigi Kronmayr

Philo Bennarsch

Joschi Bennarsch

Paul Nolte

Theo Nolte

Lydia Nolte

Gung Pahm Kien



Hans Beimer

Berta Griese

Marion Beimer


Franz Schildknecht

Henny Schildknecht

Stefan Nossek

Egon Kling

Else Kling


Elisabeth Dressler

Carsten Floeter

Vasily Sarikakis

Chris Barnsteg

Elfie Kronmayr

Sigi Kronmayr

Gottlieb Griese

Philo Bennarsch

Joschi Bennarsch

Paul Nolte

Theo Nolte

Herr Floether Der Englaender

Lydia Nolte

Gung Pahm Kien



Hans Beimer

Berta Griese

Marion Beimer


Franz Schildknecht

Henny Schildknecht

Stefan Nossek

Egon Kling

Else Kling


Elisabeth Dressler

Frank Dressler

Carsten Floeter

Vasily Sarikakis

Chris Barnsteg

Elfie Kronmayr

Sigi Kronmayr

Gottlieb Griese

Meike Schildknecht Philo Bennarsch

Joschi Bennarsch

Paul Nolte

Theo Nolte

Herr Floether Der Englaender

Figure 24: Graph A99, time points 5, 6 and 7 (first prize).

33

15070

1507315076

15079

15082

15085

15088

15091

15094

15097

15100

15103

15106

15109

15112

15115

1511815121

15124

15127

15130

15133

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15214

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15220

15223

15226

15229

1523215235

15238

15241

15244

15247

15250

15253

15256

15259

Figure

25:G

raphB

99(second

prize).

34

Figure 26: Graph B99 – general view.

35

Figure 27: Graph B99 – different views.

36

XANTHOSINE NUCLEOTIDE

RIBONUCLEIC ACIDDEOXYRIBONUCLEIC ACID

DEOXYGUANOSINE NUCLEOTIDE GUANOSINE NUCLEOTIDE

INOSINE NUCLEOTIDEADENOSINE NUCLEOTIDEDEOXYADENOSINE NUCLEOTIDE

GLUCOSE

TRIACYLGLYCEROLS ISOLEUCINE

CYTOCHROMES

URONIC ACIDS

CELLULOSE

RIBOSE 5-PNDPGLUCOSESUCROSE

FRUCTOSE

GLYCOGEN

STARCH

AMINO SUGARS

ERYTHROSE 4-P

PENTOSES

P-RIBOSYLPP

ASCORBATE

GLUCOSE 6-P

HISTIDINE

PHENYLALANINE

TYROSINE

TRYPTOPHAN

GLYCERALDEHYDE 3-P

CHORISMATE

ASPARTATE

ARGININE

LEUCINE

ALANINE

COBALAMIN

PROTOPORPHYRIN IX

CHLOROPHYLL

HEME

PLANT STEROIDS ISOPRENOIDS

PHOSPOLIPIDS

FATTY ACIDS

3-P GLYCERATE

LIGNIN

ACETYL-CoA

SUCCINATE

CITRATE

PYRUVATE

GLYCEROL

FUMARATE2-OXOGLUTERATE

OXALOACETATE

GLYOXYLATE

HORMONES

FRUCTOSE 6-P

VALINE

SERINE

GLYCINE

CHOLESTEROL

STEROID HORMONES

BILE ACIDS

GLUTAMATE

PROLINE HYDROXYPROLINE

ORNITHINE

LYSINE

THREONINE

METHIONINE

CYSTEINE

UREA

5-AMINOLEVULINATE BILE PIGMENTS

Degradation

Degradation

NAD+

CYTIDINE NUCLEOTIDE

RIBONUCLEIC ACID

DEOXYRIBONUCLEIC ACID

DEOXYCYTIDINE NUCLEOTIDE

URIDINE NUCLEOTIDE DEOXYURIDINE NUCLEOTIDE

DEOXYTHYMIDINE NUCLEOTIDE

Figure

28:G

raphC

99.

37

XANTHOSINE NUCLEOTIDE

RIBONUCLEIC ACIDDEOXYRIBONUCLEIC ACID

DEOXYGUANOSINE NUCLEOTIDE GUANOSINE NUCLEOTIDE

INOSINE NUCLEOTIDEADENOSINE NUCLEOTIDEDEOXYADENOSINE NUCLEOTIDE

GLUCOSE

TRIACYLGLYCEROLS ISOLEUCINE

CYTOCHROMES

URONIC ACIDS

CELLULOSE

RIBOSE 5-PNDPGLUCOSE

SUCROSE

FRUCTOSE

GLYCOGEN

STARCH

AMINO SUGARS

ERYTHROSE 4-P

PENTOSES

P-RIBOSYLPP

ASCORBATE

GLUCOSE 6-P

HISTIDINE

PHENYLALANINE

TYROSINE

TRYPTOPHAN

GLYCERALDEHYDE 3-P

CHORISMATE

ASPARTATE

ARGININE

LEUCINE

ALANINE

COBALAMIN

PROTOPORPHYRIN IX

CHLOROPHYLL

HEME

PLANT STEROIDS ISOPRENOIDS

PHOSPOLIPIDS

FATTY ACIDS

3-P GLYCERATE

LIGNIN

ACETYL-CoA

SUCCINATE

CITRATE

PYRUVATE

GLYCEROL

FUMARATE2-OXOGLUTERATE

OXALOACETATE

GLYOXYLATE

HORMONES

FRUCTOSE 6-P

VALINE

SERINE

GLYCINE

CHOLESTEROL

STEROID HORMONES

BILE ACIDS

GLUTAMATE

PROLINE HYDROXYPROLINE

ORNITHINE

LYSINE

THREONINE

METHIONINE

CYSTEINE

UREA

5-AMINOLEVULINATE BILE PIGMENTS

Degradation

Degradation

NAD+

CYTIDINE NUCLEOTIDE

RIBONUCLEIC ACID

DEOXYRIBONUCLEIC ACID

DEOXYCYTIDINE NUCLEOTIDE

URIDINE NUCLEOTIDE DEOXYURIDINE NUCLEOTIDE

DEOXYTHYMIDINE NUCLEOTIDE

Figure

29:G

raphC

99-

orthogonallayout.

38


In 2000 Graph Drawing Conference was helt in Colonial Williamsburg (USA) and the con-test was organised by Franz Brandenburg. Rules and data are described at:

http://www.infosun.fmi.uni-passau.de/GD2000/index.html

Graph A00

Graph A00 was proposed by M. Himsolt.First we found out that graph consists of 26 weakly connected components. After re-

moving loops from the graph we got acyclic graph. On the acyclic graph we used standardalgorithms to compute layers and position vertices into layers in order to minimize totallength of lines. Some manual repositioning of vertices was used to avoid some line cross-ings. Vertices having loops are drawn as ellipses other as rectangles. Since graph consists ofseveral components and labels are very small, some additional layouts of parts of network(top, middle and bottom) containing some components are shown.

We also produced pictures of this graph in Scalable Vector Graphics (SVG) formatbesides EPS pictures (see http://vlado.fmf.uni-lj.si/pub/gd/gd00.htm).

Graph B00

Graphs B00-A and B00-B were proposed by Ulrik Brandes.The essential part of both graphs A and B are the green vertices. But there are to many

arcs among them to produce a clear picture.We first tried with circular presentation. To reveal internal structure of green subgraph

and to determine the ordering of vertices on the circle we computed�'¡

on the green sub-graph and applied TSP (Travelling Salesman Problem) algorithm on this matrix.

�Y¡3¢ £ ¤N¥§¦¨� © ª ¢ £¦+« ª ¢ ¥§¦ ©© ª ¢ £¦+¬ ª ¢ ¥§¦ © (1)

ª ¢ ¥§¦is the neighborhood of vertex

¥®°¯:

ª ¢ ¥�¦��²±�£³³¯�´�¢ ¥®´-£¦µ>¶¸·(«

denotes the symmetric difference,¬

denotes union)Since there can exist different clusters inside the green set of vertices we extended the�H¡matrix with some additional vertices with equal distance to all green vertices. Some

vertices were repositioned manually according to connections to yellow and blue verticeson the circular net. From obtained pictures we can see that neighbouring vertices havesimilar patterns of arcs. Vertices having loops are drawn as boxes (vertices 5 and 11 ingraph A) other as circles. Different gray colors are used to show the frequency of contact:

39

� Black: 1 - weekly

� 75% Gray: 2 - biweekly

� 50% Gray: 3 - monthly

� 25% Gray: 4 - quarterly

The ordering of green vertices obtained by TSP algorithm we used also in matrix repre-sentation of graphs A and B. It seems that the matrix representation is more appropriate fordense (parts of) graphs. In the matrix representation the same shadowing is used as in thegraph layout.

The layout was awarded the first prize.The complete report of the contest is available in [9].

40

basic_fstream<char, char_traits<char>>

basic_iostream<char, char_traits<char>>basic_istream<char, char_traits<char>> basic_ios<char, char_traits<char>>

ios_base

locale

basic_ostream<char, char_traits<char>>

basic_filebuf<char, char_traits<char>>

basic_streambuf<char, char_traits<char>>

BrokenRelationException

BuildExceptionlist<string, allocator<PropertyName>>

allocator<PropertyName>

iterator

const_iterator

EntryNotFoundException

FileNotInProjectFileException

Exception

ParentScopeNotFoundException

UnimportantOperationException

UnimportantStatementException

__non_rtti_object bad_typeid

exception

codecvt<wchar_t, char, mbstate_t>

codecvt_base

facet

codecvt<char, char, state_type>

domain_error

logic_errorinvalid_argument

length_error

out_of_range

overflow_error

runtime_errorrange_error

underflow_error

failure

CppModel Model

list<Relation*, allocator<Relation*>>

allocator<Relation*>

iterator const_iterator

set<PropertyName, less<string>, allocator<string>>

allocator<string>

ProjectFile map<string, SectionContents, less<string>, allocator<SectionContents>>allocator<SectionContents>

value_compare

SectionNotFound

VariableNotFound

bad_alloc

bad_cast

bad_exception

basic_istream<wchar_t, char_traits<wchar_t>>basic_ios<wchar_t, char_traits<wchar_t>>

basic_ostream<wchar_t, char_traits<wchar_t>>

num_put<char, _Iter>

numpunct<char>

_Locimp

Array

TypeRef

Type

Entity

Element

Object

SCPos

Class

Record

CppModelFactory ModelFactory

Pointer

PointerToMemberQualifiedType

Struct

TypeDef

Union

ElementType

set<ElementType*, less<ElementType*>, allocator<ElementType*>>

allocator<ElementType*>

ILManager map<ILInstanceKey, ILInstance, less<ILInstanceKey>, allocator<ILInstance>>

allocator<ILInstance>

value_compare

Logmap<int, string, less<int>, allocator<string>>

value_compare

map<string, int, less<string>, allocator<int>>

allocator<int>

value_compare

Measurement map<string, int, less<PropertyName>, allocator<int>> value_compare

Project

set<string, less<string>, allocator<string>>

list<string, allocator<string>> iterator const_iterator

map<string, SourceFile, less<string>, allocator<SourceFile>>

allocator<SourceFile>

value_compare

_Tree<TypeRefHack, value_type, _Kfn, LessByReferencedType, allocator<TypeRefHack>>

allocator<TypeRefHack>


_Tree<TypeRefHack, value_type, _Kfn, LessByReferencedType, allocator<TypeRefHack>> iterator

_Tree<string, value_type, _Kfn, less<string>, allocator<SectionContents>> iterator const_iterator

_Tree<Cell*, value_type, _Kfn, less<Cell*>, allocator<Cell*>> allocator<Cell*>

_Tree<Cell*, value_type, _Kfn, less<Cell*>, allocator<Cell*>> iterator

const_iterator

basic_string<char, char_traits<char>, allocator<char>> allocator<char>

list<Cell*, allocator<Cell*>> iterator const_iterator

stack<Block*, deque<Block*, allocator<Block*>>> deque<Block*, allocator<Block*>> iterator const_iterator

allocator<Block*>

map<Cell*, Cell*, less<Cell*>, allocator<Cell*>> value_compare

multiset<TypeRefHack, LessByReferencedType, allocator<TypeRefHack>>

iterator

Builtin

ClassTemplate

Template

Function

Routine

FunctionTemplate

InNamespace

InScope

Relation

InRecordScope

Member

Field

Method

RoutineType

StaticDataMemberTemplate

Variable

Property

list<PropertyValue, allocator<PropertyValue>>

allocator<PropertyValue>


PropertyValue

_Tree<int, value_type, _Kfn, less<int>, allocator<string>>iterator

_Tree<int, value_type, _Kfn, less<int>, allocator<string>> iterator

_Tree<string, value_type, _Kfn, less<string>, allocator<int>>iterator const_iterator

_Tree<string, value_type, _Kfn, less<string>, allocator<int>> iterator

_Tree<RoutineTypeHack, value_type, _Kfn, LessByParamTypes, allocator<RoutineTypeHack>>

allocator<RoutineTypeHack>


_Tree<PropertyName, value_type, _Kfn, less<string>, allocator<string>>

_Tree<PropertyName, value_type, _Kfn, less<string>, allocator<string>>iterator

_Tree<ElementType*, value_type, _Kfn, less<ElementType*>, allocator<ElementType*>>

allocator<ElementType*>


_Tree<string, value_type, _Kfn, less<string>, allocator<string>> iterator

const_iterator_Tree<string, value_type, _Kfn, less<string>, allocator<string>> iterator

_Tree<string, value_type, _Kfn, less<string>, allocator<ElementType*>>

_Tree<string, value_type, _Kfn, less<string>, allocator<ElementType*>> iterator

const_iterator

_Tree<ILInstanceKey, value_type, _Kfn, less<ILInstanceKey>, allocator<ILInstance>>

_Tree<ILInstanceKey, value_type, _Kfn, less<ILInstanceKey>, allocator<ILInstance>> iterator

const_iterator

_Tree<string, value_type, _Kfn, less<ILInstanceKey>, allocator<int>>

_Tree<string, value_type, _Kfn, less<ILInstanceKey>, allocator<int>>iterator

_Tree<string, value_type, _Kfn, less<string>, allocator<string>> iteratorconst_iterator

_Tree<string, value_type, _Kfn, less<string>, allocator<string>>iterator

_Tree<string, value_type, _Kfn, less<string>, allocator<SourceFile>>allocator<SourceFile>


_Tree<ILInstanceKey, value_type, _Kfn, less<string>, allocator<ILInstance>>

_Tree<ILInstanceKey, value_type, _Kfn, less<string>, allocator<ILInstance>>iterator

_Tree<STD_Ptr<Cell>, value_type, _Kfn, CellLessByElementID, allocator<STD_Ptr<Cell>>> allocator<STD_Ptr<Cell>>

_Tree<STD_Ptr<Cell>, value_type, _Kfn, CellLessByElementID, allocator<STD_Ptr<Cell>>>iterator const_iterator

_Tree<PropertyName, value_type, _Kfn, less<PropertyName>, allocator<PropertyName>> iterator

const_iterator_Tree<PropertyName, value_type, _Kfn, less<PropertyName>, allocator<PropertyName>>

iterator

_Tree<int, value_type, _Kfn, less<int>, allocator<PropertyName>> iterator

_Tree<int, value_type, _Kfn, less<int>, allocator<PropertyName>> iterator

_Tree<string, value_type, _Kfn, less<PropertyName>, allocator<int>> iterator

_Tree<string, value_type, _Kfn, less<PropertyName>, allocator<int>> iterator

_Tree<void*, value_type, _Kfn, less<void*>, allocator<Entity*>>allocator<Entity*>

iterator

const_iterator_Tree<void*, value_type, _Kfn, less<void*>, allocator<Entity*>>iterator

_Tree<PropertyName, value_type, _Kfn, less<PropertyName>, allocator<Property>>allocator<Property>

iterator

const_iterator_Tree<PropertyName, value_type, _Kfn, less<PropertyName>, allocator<Property>>

iterator

_Tree<ILInstanceKey, value_type, _Kfn, less<PropertyName>, allocator<ILInstance>>

_Tree<ILInstanceKey, value_type, _Kfn, less<PropertyName>, allocator<ILInstance>>

iterator

_Tree<PropertyName, value_type, _Kfn, less<string>, allocator<PropertyName>>

_Tree<PropertyName, value_type, _Kfn, less<string>, allocator<PropertyName>>iterator

const_iterator

list<Entity*, allocator<Entity*>> iterator const_iterator

allocator<Entity*>

list<Element*, allocator<Element*>>

allocator<Element*>


list<TempResult, allocator<TempResult>>

allocator<TempResult>


TempResult

list<STD_Ptr<TemplateParameter>, allocator<STD_Ptr<TemplateParameter>>>

allocator<STD_Ptr<TemplateParameter>>


STD_Ptr<TemplateParameter>

list<BaseClass*, allocator<BaseClass*>>

allocator<BaseClass*>


allocator<BaseClass*>

list<Constant*, allocator<Constant*>>

allocator<Constant*>


list<Actual, allocator<Actual>>

allocator<Actual>


reverse_iterator<iterator, value_type, reference, reference*, difference_type> iteratorconst_iterator

reference

map<string, ElementType*, less<string>, allocator<ElementType*>> value_compare

map<string, int, less<ILInstanceKey>, allocator<int>> value_compare

map<string, string, less<string>, allocator<string>>value_compare

map<ILInstanceKey, ILInstance, less<string>, allocator<ILInstance>>value_compare

map<int, string, less<int>, allocator<PropertyName>> value_compare

map<void*, Entity*, less<void*>, allocator<Entity*>>value_compare

map<PropertyName, Property, less<PropertyName>, allocator<Property>>

value_compare

map<ILInstanceKey, ILInstance, less<PropertyName>, allocator<ILInstance>> value_compare

multiset<RoutineTypeHack, LessByParamTypes, allocator<RoutineTypeHack>>

set<STD_Ptr<Cell>, CellLessByElementID, allocator<STD_Ptr<Cell>>>

set<PropertyName, less<PropertyName>, allocator<PropertyName>>

set<PropertyName, less<string>, allocator<PropertyName>>

vector<_Bool, _Bool_allocator>

ConcretePrimitiveQuery

PrimitiveQuery

Query

list<ColumnDef, allocator<ColumnDef>>

allocator<ColumnDef>iterator const_iterator

ColumnDef

DummyRelation

EntitiesByTypeQuery

RelationsByTypeQuery

iterator

iterator

iterator

iterator

Accessor

Block list<STD_Ptr<Block>, allocator<STD_Ptr<Block>>>allocator<STD_Ptr<Block>>


allocator<STD_Ptr<Block>>STD_Ptr<Block>

STD_Ptr<Block>

CallFieldAccess

Friend

HasType

Inheritance

LocalVariableOfType

LocallyUsedType

NameSpace

Parameter

ReturnType

TemplateInstance

TypeUsedInTypeRef

basic_streambuf<wchar_t, char_traits<wchar_t>>

reverse_iterator<const_iterator, value_type, const_reference, const_reference*, difference_type>

ParallelQuerylist<Query*, allocator<Query*>>

allocator<Query*>iterator const_iterator

RecursiveQuery

Scopemultiset<STD_Ptr<Entity>, LessByName, allocator<STD_Ptr<Entity>>> allocator<STD_Ptr<Entity>>

set<STD_Ptr<Scope>, less<STD_Ptr<Scope>>, allocator<STD_Ptr<Scope>>> allocator<STD_Ptr<Scope>>

SerialQuery

Constant

TemplateParameter_Tree<STD_Ptr<Scope>, value_type, _Kfn, less<STD_Ptr<Scope>>, allocator<STD_Ptr<Scope>>> iterator const_iterator

_Tree<STD_Ptr<Entity>, value_type, _Kfn, LessByName, allocator<STD_Ptr<Entity>>> iterator const_iterator

Figure

30:C

omplete

layoutofH

imsoltgraph

A00.

41

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Figure 31: Graph B00-A - circular layout (first prize).

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Figure 32: Graph B00-B - circular layout.

43

9 Green 16 Green 17 Green 8 Green 11 Green 13 Green 7 Green 6 Green 2 Green 4 Green 5 Green 19 Green 62 Green 12 Green 18 Green 14 Green 1 Green 0 Green 3 Green 20 Green 15 Green 10 Green 66 Blue 76 Blue 86 Blue 92 Blue 94 Blue 67 Blue 72 Blue 80 Blue 68 Blue 82 Blue 63 Blue 73 Blue 81 Blue 100 Blue 79 Blue 97 Blue 74 Blue 77 Blue 78 Blue 91 Blue 85 Blue 71 Blue 99 Blue 90 Blue 64 Blue 95 Blue 69 Blue 70 Blue 75 Blue 83 Blue 87 Blue 89 Blue 93 Blue 96 Blue 84 Blue 65 Blue 88 Blue 98 Blue 21 Yellow 47 Yellow 33 Yellow 55 Yellow 39 Yellow 30 Yellow 57 Yellow 24 Yellow 32 Yellow 51 Yellow 48 Yellow 26 Yellow 22 Yellow 61 Yellow 49 Yellow 31 Yellow 40 Yellow 56 Yellow 25 Yellow 44 Yellow 60 Yellow 37 Yellow 41 Yellow 23 Yellow 34 Yellow 38 Yellow 45 Yellow 46 Yellow 50 Yellow 58 Yellow 52 Yellow 53 Yellow 43 Yellow 36 Yellow 54 Yellow 28 Yellow 35 Yellow 29 Yellow 59 Yellow 42 Yellow 27 Yellow

9 G

reen

1

6 G

reen

1

7 G

reen

8

Gre

en

11

Gre

en

13

Gre

en

7 G

reen

6

Gre

en

2 G

reen

4

Gre

en

5 G

reen

1

9 G

reen

6

2 G

reen

1

2 G

reen

1

8 G

reen

1

4 G

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1

Gre

en

0 G

reen

3

Gre

en

20

Gre

en

15

Gre

en

10

Gre

en

66

Blu

e 7

6 B

lue

86

Blu

e 9

2 B

lue

94

Blu

e 6

7 B

lue

72

Blu

e 8

0 B

lue

68

Blu

e 8

2 B

lue

63

Blu

e 7

3 B

lue

81

Blu

e 10

0 B

lue

79

Blu

e 9

7 B

lue

74

Blu

e 7

7 B

lue

78

Blu

e 9

1 B

lue

85

Blu

e 7

1 B

lue

99

Blu

e 9

0 B

lue

64

Blu

e 9

5 B

lue

69

Blu

e 7

0 B

lue

75

Blu

e 8

3 B

lue

87

Blu

e 8

9 B

lue

93

Blu

e 9

6 B

lue

84

Blu

e 6

5 B

lue

88

Blu

e 9

8 B

lue

21

Yel

low

47

Yel

low

33

Yel

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55

Yel

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39

Yel

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30

Yel

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Yel

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37

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41

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Yel

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46

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53

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Yel

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36

Yel

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Yel

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28

Yel

low

35

Yel

low

29

Yel

low

59

Yel

low

42

Yel

low

27

Yel

low

Figure 33: Matrix representation of graph B00-B.

44

3 Green 13 Green 11 Green 1 Green 0 Green 17 Green 9 Green 2 Green 12 Green 81 Green 5 Green 15 Green 14 Green 18 Green 10 Green 16 Green 4 Green 8 Green 7 Green 6 Green 92 Blue 91 Blue 82 Blue 83 Blue 84 Blue 85 Blue 86 Blue 87 Blue 88 Blue 93 Blue 89 Blue 96 Blue 97 Blue 98 Blue 99 Blue 90 Blue 94 Blue 95 Blue 78 Yellow 64 Yellow 25 Yellow 32 Yellow 74 Yellow 26 Yellow 27 Yellow 52 Yellow 55 Yellow 23 Yellow 60 Yellow 62 Yellow 24 Yellow 20 Yellow 77 Yellow 46 Yellow 61 Yellow 80 Yellow 28 Yellow 39 Yellow 34 Yellow 43 Yellow 72 Yellow 22 Yellow 57 Yellow 69 Yellow 70 Yellow 65 Yellow 66 Yellow 67 Yellow 31 Yellow 37 Yellow 42 Yellow 71 Yellow 19 Yellow 41 Yellow 53 Yellow 59 Yellow 73 Yellow 29 Yellow 79 Yellow 40 Yellow 21 Yellow 36 Yellow 48 Yellow 49 Yellow 50 Yellow 51 Yellow 58 Yellow 68 Yellow 44 Yellow 45 Yellow 54 Yellow 75 Yellow 33 Yellow 63 Yellow 56 Yellow 30 Yellow 76 Yellow 47 Yellow 38 Yellow 35 Yellow

3 G

reen

13

Gre

en

11 G

reen

1

Gre

en

0 G

reen

17

Gre

en

9 G

reen

2

Gre

en

12 G

reen

81

Gre

en

5 G

reen

15

Gre

en

14 G

reen

18

Gre

en

10 G

reen

16

Gre

en

4 G

reen

8

Gre

en

7 G

reen

6

Gre

en

92 B

lue

91 B

lue

82 B

lue

83 B

lue

84 B

lue

85 B

lue

86 B

lue

87 B

lue

88 B

lue

93 B

lue

89 B

lue

96 B

lue

97 B

lue

98 B

lue

99 B

lue

90 B

lue

94 B

lue

95 B

lue

78 Y

ello

w

64 Y

ello

w

25 Y

ello

w

32 Y

ello

w

74 Y

ello

w

26 Y

ello

w

27 Y

ello

w

52 Y

ello

w

55 Y

ello

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23 Y

ello

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60 Y

ello

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62 Y

ello

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24 Y

ello

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20 Y

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77 Y

ello

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46 Y

ello

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61 Y

ello

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80 Y

ello

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28 Y

ello

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39 Y

ello

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34 Y

ello

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43 Y

ello

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72 Y

ello

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22 Y

ello

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57 Y

ello

w

69 Y

ello

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70 Y

ello

w

65 Y

ello

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66 Y

ello

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67 Y

ello

w

31 Y

ello

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37 Y

ello

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42 Y

ello

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71 Y

ello

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19 Y

ello

w

41 Y

ello

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53 Y

ello

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59 Y

ello

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73 Y

ello

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29 Y

ello

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79 Y

ello

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40 Y

ello

w

21 Y

ello

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36 Y

ello

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48 Y

ello

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49 Y

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50 Y

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51 Y

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58 Y

ello

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68 Y

ello

w

44 Y

ello

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45 Y

ello

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54 Y

ello

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75 Y

ello

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33 Y

ello

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63 Y

ello

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56 Y

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30 Y

ello

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76 Y

ello

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47 Y

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38 Y

ello

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35 Y

ello

w

9 Green 16 Green 17 Green 8 Green 11 Green 13 Green 7 Green 6 Green 2 Green 4 Green 5 Green 19 Green 62 Green 12 Green 18 Green 14 Green 1 Green 0 Green 3 Green 20 Green 15 Green 10 Green 66 Blue 76 Blue 86 Blue 92 Blue 94 Blue 67 Blue 72 Blue 80 Blue 68 Blue 82 Blue 63 Blue 73 Blue 81 Blue 100 Blue 79 Blue 97 Blue 74 Blue 77 Blue 78 Blue 91 Blue 85 Blue 71 Blue 99 Blue 90 Blue 64 Blue 95 Blue 69 Blue 70 Blue 75 Blue 83 Blue 87 Blue 89 Blue 93 Blue 96 Blue 84 Blue 65 Blue 88 Blue 98 Blue 21 Yellow 47 Yellow 33 Yellow 55 Yellow 39 Yellow 30 Yellow 57 Yellow 24 Yellow 32 Yellow 51 Yellow 48 Yellow 26 Yellow 22 Yellow 61 Yellow 49 Yellow 31 Yellow 40 Yellow 56 Yellow 25 Yellow 44 Yellow 60 Yellow 37 Yellow 41 Yellow 23 Yellow 34 Yellow 38 Yellow 45 Yellow 46 Yellow 50 Yellow 58 Yellow 52 Yellow 53 Yellow 43 Yellow 36 Yellow 54 Yellow 28 Yellow 35 Yellow 29 Yellow 59 Yellow 42 Yellow 27 Yellow

9 G

reen

1

6 G

reen

1

7 G

reen

8

Gre

en

11

Gre

en

13

Gre

en

7 G

reen

6

Gre

en

2 G

reen

4

Gre

en

5 G

reen

1

9 G

reen

6

2 G

reen

1

2 G

reen

1

8 G

reen

1

4 G

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1

Gre

en

0 G

reen

3

Gre

en

20

Gre

en

15

Gre

en

10

Gre

en

66

Blu

e 7

6 B

lue

86

Blu

e 9

2 B

lue

94

Blu

e 6

7 B

lue

72

Blu

e 8

0 B

lue

68

Blu

e 8

2 B

lue

63

Blu

e 7

3 B

lue

81

Blu

e 10

0 B

lue

79

Blu

e 9

7 B

lue

74

Blu

e 7

7 B

lue

78

Blu

e 9

1 B

lue

85

Blu

e 7

1 B

lue

99

Blu

e 9

0 B

lue

64

Blu

e 9

5 B

lue

69

Blu

e 7

0 B

lue

75

Blu

e 8

3 B

lue

87

Blu

e 8

9 B

lue

93

Blu

e 9

6 B

lue

84

Blu

e 6

5 B

lue

88

Blu

e 9

8 B

lue

21

Yel

low

47

Yel

low

33

Yel

low

55

Yel

low

39

Yel

low

30

Yel

low

57

Yel

low

24

Yel

low

32

Yel

low

51

Yel

low

48

Yel

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26

Yel

low

22

Yel

low

61

Yel

low

49

Yel

low

31

Yel

low

40

Yel

low

56

Yel

low

25

Yel

low

44

Yel

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60

Yel

low

37

Yel

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41

Yel

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23

Yel

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34

Yel

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Yel

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Yel

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Yel

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Yel

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58

Yel

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Yel

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53

Yel

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Yel

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36

Yel

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54

Yel

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28

Yel

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35

Yel

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29

Yel

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59

Yel

low

42

Yel

low

27

Yel

low

Figure

34:M

atrixrepresentation

ofnon

empty

partofgraphs

B00-A

andB

00-B.

45


In 2001 Graph Drawing Conference was helt in Vienna and the contest was organised byFranz Brandenburg. Rules and data are described at:

http://www.ads.tuwien.ac.at/gd2001/

Graph A01

In a given set of units¶

(articles, books, works, . . . ) we introduce a ¹�ºT»bºT¼k½ relation ¾À¿¶ÂÁÃ¶ Ä¾ÆÅ�ÇÈÅ cites

Äwhich determines a citation network

¢É¶�¤ ¾ ¦ . A citing relation is usually irreflexive, ÊÄ ¶Ë´�Ì Ä ¾

Ä, and (almost) acyclic, Ê

Ä @¶ ÊÍ IN� ´�Ì Ä ¾8Î

Ä. The citation network is

standardized by adding, if necessary, artificial source vertex � and sink or terminal vertex »and the arc

¢ » ¤ � ¦ (see figure).

An approach to the analysis of citation network is to determine for each unit / arc itsimportance or weight. These values are used afterwards to determine the essential substruc-tures in the network. Hummon and Doreian (1989, 1990) [1, 2, 3] proposed three methodsof assigning weights Ï ´ ¾@Ð IR

�Ñ to arcs:

46

� NPCC method: Ï � ¢ £Ò¤N¥�¦�� © ¾ÔÓ � ¢ £¦ ©1ÕH© ¾ ¢ ¥§¦ ©� Paths count method: Ï ¡3¢ £Ò¤N¥�¦Ö� ª ¢ £Ò¤N¥§¦, where ª ¢ £Ò¤N¥�¦

denotes the number ofdifferent paths from ×³ØoÙ�¾ to ×³Ú1ÛÜ¾ (or from � to » ) through the arc

¢ £Ò¤N¥�¦� SPLC method: ÏÞÝ ¢ £ ¤N¥§¦ß� ªÖà ¢ £Ò¤N¥§¦ , where ªÖà ¢ £Ò¤N¥§¦ equals to ª ¢ £Ò¤N¥�¦

from pathscount method over the network

¢É¶�¤ ¾ à ¦ , ¾ à ´��¢ ¾ ¬á± � ·1¦âÁ�¢É¶äãK± � ¤ » ·1¦The last two methods are efficiently (Batagelj, 1991, 1994) [4] implemented in Pajek

and can be applied also on (very) large acyclic networks. Let ª Ó ¢ ¥�¦ denotes the numberof different paths from � to

¥, and ª � ¢ ¥§¦ denotes the number of different paths from

¥to» . Then ª ¢ £Ò¤N¥�¦u� ª Ó ¢ £¦ Õ�ª � ¢ ¥�¦:¤�¢ £Ò¤N¥§¦Ü ¾ .ª and ªÖà are flows in the network since they obey the Kirchoff’s node law:

For every node¥

in a citation network¢É¶�¤ ¾ ¦ in standard form it holds

incoming flow�

outgoing flow

Therefore the total flow through the citation network equals ª ¢ » ¤ � ¦ . This gives us a naturalway to normalize the weights

Ï ¢ £Ò¤N¥�¦u� ª ¢ £Ò¤N¥�¦ª ¢ » ¤ � ¦ å æ�ç Ï ¢ £ ¤N¥§¦ ç²è

If é is a minimal cut-set it also holds êë�ì-í î:ïZð�ñ Ï ¢ £Ò¤N¥�¦�� è

We can assign weights also to vertices

Ï ¢ ¥�¦u�áª Ó ¢ ¥§¦ Õ�ª � ¢ ¥§¦ª ¢ » ¤ � ¦References

Layouts of graph A

In graph A the relation is the reverse of the citing relation. The original graph A has 311vertices. It has 6 weak components. Searching for strong components (testing acyclicity)it turns out that the graph A is not acyclic. A large strong component was generated byan erroneous arc ( GD94/143 Eades, GD98/423 Eades). After reversing it 4 small strongcomponents remained, corresponding to mutual references

±GD94/286 Garg, GD94/298

Papakostas·,±

GD94/328 Di Battista, GD94/340 Bose, GD94/352 ElGindy·,±

GD95/8Alt, GD95/234 Fekete

·and

±GD95/140 Chandramouli, GD95/300 Heath

·. To obtain

an acyclic graph, required by citations analysis method, we applied the following ’preprint’transformation:

47

Each paper from a strong component is duplicated with its ’preprint’ version. The papersinside strong component cite preprints.

The pictures were exported as nested partitions into SVG format that allows interactivedisplay of different slices – subgraphs induced by arcs with weights larger than a thresholdvalue. These pictures are available at

http://vlado.fmf.uni-lj.si/pub/GD/GD01.htm.In the paper form we can present only some snapshots.The first picture displays complete network after ’preprint’ transformations (320 ver-

tices). The vertices are put in layers (vertical position and color of vertices) according tothe years of publication. The placement of vertices inside the layer was determined by lo-cal optimization. The width and the color density of an arc and the size of a vertex areproportional to their citation weights.

The second and the third picture display the main parts (slices) of the citation networkat threshold values 0.02 and 0.05. The red arcs belong to the ’main path’.

48

Figure 35: Graph A – complete graph.

49

Figure 36: Graph A – level 0.02.

50

Figure 37: Graph A – level 0.05.

51

Graph B

To obtain the ’central symmetric’ picture of graph B energy drawing was used, followedby manual grid positioning of vertices. To save the space the lower part of the picture wasmanually mirrored across the vertical axis.

P0

T8+

P-2

2

P0

P-22

T8+

P+

22

T9+

P-2

2

T9+

T9+

T8-

T8+

T9+

T9-

T8-

T8+

T9+

T9-

T8-

T8+

T9-T

9+

T8-

T9-

T9+

T0+

T9+

P-90

T9+ T8+

P-6

0

P+90

T9+

T8+T8+

T8+

+90 posturn right

turn right equalizer

0 pos turn left equalizer

+22 pos

turn left+22 - +90 intermediate

0 - +22 intermediate


-22 pos

turn right

-90 pos turn left

-60 pos

turn left equalizer

0 - -22 intermediate

-22 - -90 intermediate

Figure 38: Graph B – ’central symmetric’.

Graph C

Graph C is an acyclic directed graph. Such graphs can be topologically sorted. The corre-sponding adjacency matrix has zero lower triangle and diagonal. Since the graph is ratherdense we decided to use the matrix representation to visualize the graph structure. Layersare represented by blocks divided by blue lines.

52

P0

T8+

P-2

2P

0P

-22

T8+

P+

22T

9+P

-22

T9+

T9+

T8-

T8+

T9+

T9-

T8-

T8+

T9+

T9-

T8-

T8+

T9-T

9+

T8-

T9-

T9+

T0+

T9+

P-90

T9+

T8+ P

-60

P+90

T9+

T8+T8+

T8+

+90 posturn right


0 posturn left equalizer

+22 pos

turn left+22 - +90 intermediate

0 - +22 intermediate


-22 pos

turn right

-90 posturn left

-60 pos

turn left equalizer

0 - -22 intermediate

-22 - -90 intermediate

Figure 39: Graph B – mirror.

53

Pajek - shadow [0.00,1.00]

RGCLGNV1V2V3VPPIPV3AMDPMIPPOMTV4tV4VOTVIPLIPMSTFSTPIT7aFEFSTPpCIT46STPaAITOFC36TFTHERHC

RG

CLG

NV

1V

2V

3V

PP

IPV

3AM

DP

MIP

PO

MT

V4t

V4

VO

TV

IPLI

PM

ST

FS

TP

IT7a F

EF

ST

Pp

CIT

46 ST

Pa

AIT

OF

C36 T

FT

HE

RH

C

Figure 40: Graph C – matrix representation.

REFERENCES 54

References

[1] Hummon N.P., Doreian P. (1989): Connectivity in a Citation Network: The Develop-ment of DNA Theory. Social Networks, 11 39-63.

[2] Hummon N.P., Doreian P. (1990): Computational Methods for Social Network Anal-ysis. Social Networks, 12 273-288.

[3] Hummon N.P., Doreian P., Freeman L.C. (1990): Analyzing the Structure of theCentrality-Productivity Literature Created Between 1948 and 1979. Knowledge: Cre-ation, Diffusion, Utilization, 11(4), 459-480.

[4] Batagelj V. (1991): An Efficient Algorithm for Citation Networks Analysis. Presentedat EASST’94, Budapest, Hungary, August 28-31, 1994. First presented at the Seminaron social networks. University of Pittsburgh, January 1991.

[5] Batagelj V., Doreian P., and Ferligoj A. (1992): An Optimizational Approach to Regu-lar Equivalence. Social Networks 14, 121-135.

[6] Brandenburg, F. J. (1996): Graph Drawing. Proceedings of Symposium on GraphDrawing, GD95, Passau, Germany, September 1995, Springer-Verlag, Lecture Notesin Computer Science, vol. 1027, 224-233.

[7] DiBattista, G. (1998): Graph Drawing. Proceedings of Symposium on Graph Draw-ing, GD97, Rome, Italy, September 1997, Springer-Verlag, Lecture Notes in ComputerScience, vol. 1353, 438-445.

[8] Kratochvil, J. (1999): Graph Drawing. Proceedings of Symposium on Graph Draw-ing, GD99, Stirin Castle, Czeck Republic. September 15-19, 1999. Springer-Verlag,Lecture Notes in Computer Science, vol. 1731, 400-409.

[9] Marks, J. (2001): Graph Drawing. Proceedings of the 8th Symposium on Graph Draw-ing, GD00, Colonial Williamsburg, USA. September 20-23, 2000. Springer-Verlag,Lecture Notes in Computer Science, vol. 1984, 410-418.

[10] North, S. (1997): Graph Drawing. Proceedings of Symposium on Graph Drawing,GD96, Berkeley, California, USA, September 1996, Springer-Verlag, Lecture Notesin Computer Science, vol. 1190, 129-138.

layouts for graph drawing contests 1995-2001

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