UNIVERSITY OF LJUBLJANA I NSTITUTE OF MATHEMATICS,PHYSICS AND MECHANICS DEPARTMENT OF THEORETICAL COMPUTER SCIENCE JADRANSKA 19, 1 000 LJUBLJANA,SLOVENIA Preprint series, Vol. 40 (2002), 568/2 LAYOUTS FOR GRAPH DRAWING CONTESTS 1995-2001 Vladimir Batagelj, Andrej Mrvar ISSN 1318-4865 Version 1: May 13, 1998 Version 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, Department of Mathematics, and IMFM Ljubljana, Department of TCS, Jadranska ulica 19, 1 000 Ljubljana, Slovenia e-mail: [email protected]Ljubljana, December 27, 2001
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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
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 pictures in some (dynamical) graphical formats (VRML, SVG) thatcan not be adequately reproduced in the paper form.
2
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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MUX1
INSTRUCTION BUFFER (4 x 16)2
RCV3
DRV4
ROM REG5
MUX6
ROM ADDR7
MACRO INSTRUCTION ROM8
MUX9
L210
MUX11
GENERAL REGISTER FILE (32 x 32)12
ALU13
A STAGE14
MUX15
L116
MUX17
SHIFTER18
MUX19
J REG20
PROGRAM COUNTER21
PSW SSW22
B STAGE23
C STAGE24
PIPELINE CONTROL25
OUTPUT26
ALIGN27
INPUT28
MUX29
R30
ALU31
ACCUMULATOR32
FPU REGISTER FILE (8 X 64)33
ENCODER34
INSTRUCTION BUS TO I-CAMMU35
DATA BUS TO D-CAMMU36
INSTRUCTION BUS INTERFACE
MACRO INSTRUCTION UNIT
INTEGER EXECUTION UNIT
INSTRUCTION CONTROL UNIT
DATA BUS INTERFACE
FLOATING POINT UNIT
EXTERNAL
Figure 2: Graph A95 descriptions.
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Figure 3: Graph B95 (honorable mention).
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graphics.c:GRDisplayPoint1
long1.c:GRDisplayPoint2
long2.c:GRDisplayPoint3
short.c:GRDisplayPoint4
X11/X.h:GCFont5
gremlin.h:dby_to_win6
X11/X.h:GXxor7
X11/X.h:GCFunction8
X11/Xlib.h:XGCValues9
gremlin.h:dbx_to_win10
graphics.c:text_pf11
main.c:pix_gc12
main.c:display13
X11/Xlib.h:XChangeGC14
X11/Xlib.h:XCopyPlane15
/usr/include/strings.h:strlen16
main.c:Artmode17
main.c:SUN_YORIGIN18
main.c:SUN_XORIGIN19
main.c:pix_sw20
X11/Xlib.h:XDrawString21
/usr/include/stdio.h:sprintf22
icons/icon.littlepoint:littlepoint_pm23
X11/Xlib.h:struct Xlib_h_324
X11/Xlib.h:XFontStruct25
icondata.c:mpr_static26
gremlin.h:PMRec27
X11/Xlib.h:Display28
gremlin.h:FALSE29
X11/X.h:Window30
X11/Xlib.h:GC31
gremlin.h:struct PMRec32
X11/X.h:Font33
X11/Xlib.h:struct _XDisplay34
X11/Xlib.h:Bool35
X11/Xlib.h:struct _XGC36
X11/X.h:Pixmap37
X11/X.h:XID38
X11/Xlib.h:struct Xlib_h_5739
X11/Xlib.h:ScreenFormat40
X11/Xlib.h:struct Xlib_h_2241
X11/Xlib.h:Status42
X11/Xlib.h:struct Xlib_h_2143
X11/Xlib.h:XCharStruct44
X11/Xlib.h:XExtData45
X11/X.h:KeySym46
X11/X.h:GContext47
X11/Xlib.h:Screen48
X11/Xlib.h:XModifierKeymap49
X11/Xlib.h:XFontProp50
<void>:struct _XrmHashBucketRec51
<void>:struct _XKeytrans52
<void>:struct _XIMFilter53
<void>:struct _XExten54
<void>:struct _XContextDB55
<void>:struct _XFreeFuncs56
<void>:struct _XSQEvent57
<void>:struct _XDisplayAtoms58
X11/Xlib.h:struct Xlib_h_2059
X11/Xlib.h:struct Xlib_h_760
X11/Xlib.h:struct _XExtData61
X11/Xlib.h:XPointer62
X11/Xlib.h:struct Xlib_h_663
X11/Xlib.h:struct Xlib_h_5664
X11/Xlib.h:struct Xlib_h_5565
X11/X.h:Atom66
X11/Xlib.h:Depth67
X11/X.h:Colormap68
X11/X.h:KeyCode69
X11/Xlib.h:Visual70
X11/Xlib.h:struct Xlib_h_571
X11/Xlib.h:struct Xlib_h_472
X11/X.h:VisualID73
procedure
macro
type
variable
Figure 4: Graph B95 descriptions.
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Figure 5: Graph C95 (first prize).
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t-node
o-node
Figure 6: Graph C95 descriptions.
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Layouts for Graph Drawing Contest 1996
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:
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.
The layout was awarded the first prize.
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 7: 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
11
207-555
415-555
303-555
617-555
313-555
406-555
304-555
603-555
Figure 8: Graph B96 types.
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207-555-82481
415-555-21882
415-555-60333
303-555-43934
617-555-97655
313-555-03296
415-555-79627
313-555-85638
207-555-18039
406-555-083610
303-555-795511
617-555-926012
415-555-239813
617-555-099114
406-555-590315
207-555-424816
406-555-467617
415-555-032318
207-555-865919
313-555-385320
304-555-848421
617-555-329022
303-555-987823
415-555-468924
415-555-631725
313-555-012226
207-555-064827
303-555-683328
617-555-143629
303-555-156530
313-555-290631
415-555-826732
313-555-258333
617-555-963034
415-555-333035
207-555-780736
415-555-862737
303-555-454338
603-555-814839
304-555-017140
617-555-992541
313-555-924342
207-555-074043
415-555-240144
406-555-893045
207-555-054146
303-555-035747
313-555-475348
617-555-613349
415-555-606250
617-555-041451
303-555-270752
313-555-400553
415-555-059054
207-555-266955
617-555-698856
303-555-630057
617-555-554258
415-555-526659
304-555-949660
603-555-432161
303-555-642962
313-555-135663
207-555-863564
207-555-071065
617-555-557566
313-555-077867
603-555-656568
313-555-500769
304-555-516770
415-555-590871
207-555-339472
617-555-920673
303-555-922374
415-555-760275
415-555-119776
304-555-466077
207-555-167678
406-555-128579
303-555-276980
303-555-027581
303-555-648182
617-555-625783
313-555-297884
415-555-881285
313-555-051586
207-555-852087
303-555-718988
617-555-473489
313-555-526890
415-555-527991
603-555-121392
207-555-982293
303-555-038694
304-555-566195
207-555-728196
415-555-835597
303-555-857698
304-555-235499
617-555-1958100
207-555-5013101
313-555-7592102
415-555-5058103
617-555-9344104
313-555-4790105
313-555-7556106
603-555-0525107
406-555-5703108
406-555-8517109
415-555-6423110
603-555-5694111
Figure 9: Graph B96 labels.
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Figure10:G
raphC
96(firstprize).
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Figure 11: Graph C96 / VRML snapshot.
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Figure 12: Graph C96 with cylinder / VRML snapshot.
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.
The layout was awarded the first prize.
� 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.
The layout was awarded the first prize.
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
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 us-ing macro language which is recognized by program package Pajek. Then we usedMicrosoft 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.
The layout was awarded the first prize.
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.
27
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
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 short-est paths between graph and drawing and between drawing and contest, drawthe obtained graph using layers option in Pajek (layers are determined by distance fromdrawing). Additionally we added two puzzles:� one difficult: find shortest paths alone,
28
Figure 22: Graph C98.
� one easy: find only missing words on the paths.
See also: http://vlado.fmf.uni-lj.si/pub/gd/gd98.htm
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:
– 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)
37
– =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.
The layout was awarded the first prize.
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].
38
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+18.08.1996Figure30:G
raphA
99,currentsituation.
39
Lydia Nolte
Gung Pahm Kien
Gabi Zenker-geb.Skawowski
Helga Beimer Schiller
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
Gabi Zenker-geb.Skawowski
Helga Beimer Schiller
Hans Beimer
Berta Griese
Marion Beimer
Benny BeimerKlaus Beimer
Franz Schildknecht
Henny Schildknecht
Stefan Nossek
Egon Kling
Else Kling
Tanja Schildknecht-DresslerDr. Ludwig Dressler
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
Gabi Zenker-geb.Skawowski
Helga Beimer Schiller
Hans Beimer
Berta Griese
Marion Beimer
Benny BeimerKlaus Beimer
Franz Schildknecht
Henny Schildknecht
Stefan Nossek
Egon Kling
Else Kling
Tanja Schildknecht-DresslerDr. Ludwig Dressler
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 31: Graph A99, time points 5, 6 and 7 (first prize).
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:
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.
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:
46
� 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].
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
w
23 Y
ello
w
60 Y
ello
w
62 Y
ello
w
24 Y
ello
w
20 Y
ello
w
77 Y
ello
w
46 Y
ello
w
61 Y
ello
w
80 Y
ello
w
28 Y
ello
w
39 Y
ello
w
34 Y
ello
w
43 Y
ello
w
72 Y
ello
w
22 Y
ello
w
57 Y
ello
w
69 Y
ello
w
70 Y
ello
w
65 Y
ello
w
66 Y
ello
w
67 Y
ello
w
31 Y
ello
w
37 Y
ello
w
42 Y
ello
w
71 Y
ello
w
19 Y
ello
w
41 Y
ello
w
53 Y
ello
w
59 Y
ello
w
73 Y
ello
w
29 Y
ello
w
79 Y
ello
w
40 Y
ello
w
21 Y
ello
w
36 Y
ello
w
48 Y
ello
w
49 Y
ello
w
50 Y
ello
w
51 Y
ello
w
58 Y
ello
w
68 Y
ello
w
44 Y
ello
w
45 Y
ello
w
54 Y
ello
w
75 Y
ello
w
33 Y
ello
w
63 Y
ello
w
56 Y
ello
w
30 Y
ello
w
76 Y
ello
w
47 Y
ello
w
38 Y
ello
w
35 Y
ello
w
Figure 43: Matrix representation of graph B00-A.
54
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
reen
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
low
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
low
60
Yel
low
37
Yel
low
41
Yel
low
23
Yel
low
34
Yel
low
38
Yel
low
45
Yel
low
46
Yel
low
50
Yel
low
58
Yel
low
52
Yel
low
53
Yel
low
43
Yel
low
36
Yel
low
54
Yel
low
28
Yel
low
35
Yel
low
29
Yel
low
59
Yel
low
42
Yel
low
27
Yel
low
Figure 44: Matrix representation of graph B00-B.
55
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
w
23 Y
ello
w
60 Y
ello
w
62 Y
ello
w
24 Y
ello
w
20 Y
ello
w
77 Y
ello
w
46 Y
ello
w
61 Y
ello
w
80 Y
ello
w
28 Y
ello
w
39 Y
ello
w
34 Y
ello
w
43 Y
ello
w
72 Y
ello
w
22 Y
ello
w
57 Y
ello
w
69 Y
ello
w
70 Y
ello
w
65 Y
ello
w
66 Y
ello
w
67 Y
ello
w
31 Y
ello
w
37 Y
ello
w
42 Y
ello
w
71 Y
ello
w
19 Y
ello
w
41 Y
ello
w
53 Y
ello
w
59 Y
ello
w
73 Y
ello
w
29 Y
ello
w
79 Y
ello
w
40 Y
ello
w
21 Y
ello
w
36 Y
ello
w
48 Y
ello
w
49 Y
ello
w
50 Y
ello
w
51 Y
ello
w
58 Y
ello
w
68 Y
ello
w
44 Y
ello
w
45 Y
ello
w
54 Y
ello
w
75 Y
ello
w
33 Y
ello
w
63 Y
ello
w
56 Y
ello
w
30 Y
ello
w
76 Y
ello
w
47 Y
ello
w
38 Y
ello
w
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
reen
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
low
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
low
60
Yel
low
37
Yel
low
41
Yel
low
23
Yel
low
34
Yel
low
38
Yel
low
45
Yel
low
46
Yel
low
50
Yel
low
58
Yel
low
52
Yel
low
53
Yel
low
43
Yel
low
36
Yel
low
54
Yel
low
28
Yel
low
35
Yel
low
29
Yel
low
59
Yel
low
42
Yel
low
27
Yel
low
Figure45:M
atrixrepresentation
ofnonem
ptypartofgraphs
B00-A
andB
00-B.
56
Layouts for Graph Drawing Contest 2001
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¿¶ÂÁö ľÆÅ�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
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:
58
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’.
59
Figure 46: Graph A – complete graph.
60
Figure 47: Graph A – level 0.02.
61
Figure 48: Graph A – level 0.05.
62
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-T9+
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
turn right equalizer
-22 pos
turn right
-90 pos turn left
-60 pos
turn left equalizer
0 - -22 intermediate
-22 - -90 intermediate
Figure 49: 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.
[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.