Mathematical Aspects ofPublic Transportation Networks
Niels Lindner
July 9, 2018
July 9, 2018 1 / 48
Reminder
Dates
I July 12: no tutorial (excursion)
I July 16: evaluation of Problem Set 10 and Test Exam
I July 19: no tutorial
I July 19/20: block seminar on shortest paths
I August 7 (Tuesday): 1st exam
I October 8 (Monday): 2nd exam
Exams
I location: ZIB seminar room
I start time: 10.00am
I duration: 60 minutes
I permitted aids: one A4 sheet with notes
July 9, 2018 2 / 48
Chapter 6
Metro Map Drawing
§6.1 Metro Maps
July 9, 2018 3 / 48
§6.1 Metro Maps
Berlin, 1921 (Pharus-Plan)
alt-berlin.info
July 9, 2018 4 / 48
§6.1 Metro Maps
Berlin, 1913 (Hochbahngesellschaft)
Sammlung Mauruszat, u-bahn-archiv.de
July 9, 2018 5 / 48
§6.1 Metro Maps
Berlin, 1933 (BVG)
Sammlung Mauruszat, u-bahn-archiv.de
July 9, 2018 6 / 48
§6.1 Metro Maps
London, 1926 (London Underground)
commons.wikimedia.org
July 9, 2018 7 / 48
§6.1 Metro Maps
London, 1933 (Harry Beck)
July 9, 2018 8 / 48
§6.1 Metro Maps
Plaque at Finchley Central station
Nick Cooper, commons.wikimedia.org
July 9, 2018 9 / 48
§6.1 Metro Maps
Berlin, 1968 (BVG-West)
Sammlung Mauruszat, u-bahn-archiv.de
July 9, 2018 10 / 48
§6.1 Metro Maps
Berlin, 2018 (BVG)
RB55
RE5 RB12
RB20RB27
RE3 RB24
RB25
RB26
RE1
RB36
RE4 RE5RE7 RB33
RB23
RE1
RB20 RB22
RE4
RB13RB21
RE2
RB12 RB25
RB26
RB27
RB27
RB27
RB27
RB14
RE3 RE7
RB14
RE1
RE1
RE2RB24
RE6. RB20
RB20 RE5.RB12
RB12 .RB24RE3
. RE5. RE6
6. RB27
RE4.RE6.RB10.RB13
RE4.RE6.RB10
RB20
. RB2
1
RB23
RB22
RE1. RE7
. RB21
RB22. RB33
RE5.RB22 RE7.RB22
RE7. RB14
. RB22
RE2 .RB22 .RB24
RE2.
RE7.
RB14
RE1
RE3. RE4
. RE5
RE3.RE5.RE6
RE3.
RE4.
RE5
RB10
RB12
. RB2
4.RB
25
RB25
RB26
RE2.
RB14
RE2 .RE6 .RB10 .RB14
RE4.RB13
RB27
RE2.RE4.RE6.RB10.RB13.RB14
RE3 .RE4 .RE5 .RB10
RB36
RB20
RB23
RB20
. RB2
1.RB
22
RB22
RB21
RE1.RE2.RE7.RB14
RB10
RB33
RE1. RE7
. RB21. RB22
RB21 RB22
RE6 RB55
RB10
RE6 RE66
RB22 RB36
RB24
RB24
RB13
RE1 .RE2 .RE7 .RB14
RE66
1
1 3
3
2
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4
4
:5
:5
eE
eE
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bE
bE
bF
bF
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<S41
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<S41
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<S42
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CBA
X7 171 N7
X9
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TXL 128
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128
TXL
X9
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Schönefeld SXF0A
0A Tegel TXL
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<
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<
<
<
Velten (Mark)
Bärenklau
Vehlefanz
Templin Stadt Stralsund/Rostock
Hohen Neuendorf West Schönwalde (Barnim)
Schönerlinde
Basdorf
Wandlitz
Wandlitzsee
Zühlsdorf
Wensickendorf
Schmachtenhagen
Groß Schönebeck (Schorfheide)
Rüdnitz
Stralsund/Schwedt (Oder) Szczecin (Stettin) Eberswalde
Werneuchen
Blumberg-Rehhahn
Blumberg
Ahrensfelde Nord
Kostrzyn(Küstrin)
Fangschleuse
Frankfurt(Oder)
Eisen-hütten-
stadt
Niederlehme ZernsdorfFrankfurt
(Oder)
Cottbus
Senftenberg
Dahlewitz
Rangsdorf
Elsterwerda Wünsdorf-Waldstadt
Thyrow
Jüterbog
Ludwigsfelde Struveshof
Medienstadt Babelsberg
Rehbrücke
Wilhelmshorst
Saarmund
Jüterbog
Michendorf
SeddinFerch-
Lienewitz
CaputhSchwielowsee
Caputh-Geltow
Pirschheide
ParkSans-souci
Char-lotten-
hof
Werder(Havel)
Branden-burg
Magde-burg
Golm
Marquardt
Priort
Wuster-mark
Rathenow
Brieselang
Wismar
Nauen
ElstalDallgow-Döberitz
Staaken
Albrechtshof
Seegefeld
Falkensee
Finkenkrug
Ludwigsfelde
Ahrensfelde Friedhof
Teltow
Großbeeren
Birkengrund
Lutherstadt Wittenberg/Falkenberg (Elster)
Kablow
Dessau
Zeesen
Kremmen Wittenberge
Seefeld (Mark)
Sachsenhausen (Nordb)
Rathaus Spandau
Rathaus Steglitz
Alt-Mariendorf
Wittenau
Osloer Str.
Pankow
Schöneweide
Uhlandstr.
Spandau
Strausberg Nord
Hermannstr.
Krumme Lanke
Potsdam Hbf
Alt-Tegel
Hennigsdorf
Oranienburg
Wartenberg
Ahrensfelde
ErknerSpindlersfeld
Hönow
Flughafen Berlin-Schönefeld
Königs Wusterhausen
Innsbrucker Platz
Bernau
Teltow Stadt
Blankenfelde
Grünau
Zeuthen
WaidmannslustKarow
Westkreuz Charlotten-burg
Hohen Neuendorf
Blankenburg
Treptower Park
Lichtenberg
Ostkreuz
HauptbahnhofWestend Springpfuhl
Südkreuz
Wannsee
Nollen-dorfplatz
Warschauer Str.
Ruhleben
Bundesplatz
Alexanderplatz
Grunewald
Walther-Schreiber-Platz
Britz-Süd
Hackescher Markt
Schwartz-kopffstr.
Köpenick
Oskar-Helene-Heim
Mehringdamm Hermannplatz
Lichtenrade
Rudow
Ostbahnhof
Lichterfelde Süd
HalemwegSiemens-
dammRohrdammPaulsternstr.HaselhorstZitadelle
Altstadt Spandau
Birkenstr.
TiergartenErnst-Reuter-
PlatzDeutsche
Oper
Richard-Wagner-Platz
Bismarckstr.Sophie-
Charlotte-Platz
Savignyplatz
Olympia-Stadion
Neu-Westend
Theodor-Heuss-Platz
MesseSüd
Messe Nord/ICCHeerstr.
Olympiastadion
Pichelsberg
Stresow
Konstanzer Str.
Fehrbelliner Platz
Spichernstr.
Blissestr.
Berliner Str.
Heidelberger PlatzRüdesheimer Platz
Breitenbachplatz
Podbielskiallee
Dahlem-Dorf
Freie UniversitätThielplatz
Onkel Toms Hütte
Friedrich-Wilhelm-Platz
Zehlendorf
Mexikoplatz
Sundgauer Str.
Botanischer GartenSchlachtensee
Babelsberg
Attilastr.Südende
Schichauweg
Gneisenaustr. Südstern
Boddinstr.
Leinestr.
Rathaus Neukölln
Karl-Marx-Str.
Neukölln
Grenzallee
Parchimer Allee
Platz der Luftbrücke
Paradestr.
Tempelhof
Kaiserin-Augusta-Str.
Ullsteinstr.
Westphalweg
Lipschitzallee
Wutzkyallee
Zwickauer Damm
Schlesisches Tor
Heinrich-Heine-Str.
Schönleinstr.
Prinzenstr.
Weberwiese
Strausberger PlatzSchillingstr.
MärkischesMuseum
Klosterstr.
Spittel-markt
HermsdorfFrohnau
BirkenwerderBorgsdorf
Lehnitz
Wilhelmsruh
SchönholzPankow-Heinersdorf
Buch
RöntgentalZepernick
Bernau-Friedenstal
Reinicken-dorfer Str. Humboldthain
Voltastr.
Leopoldplatz
Nauener Platz
Afrikanische Str. Franz-Neumann-PlatzAm Schäfersee
Residenzstr.
Paracelsus-Bad
Lindauer Allee
Alt-Reinickendorf
Amrumer Str.
Westhafen
Scharnweberstr.
Otisstr.Holzhauser Str.
Borsigwerke
Rathaus Reinickendorf
Eichborndamm
Tegel
Schulzendorf
Heiligensee
Gehrenseestr.
Mehrower Allee
Raoul-Wallenberg-Str.
Betriebsbahnhof Rummelsburg
Rummelsburg
Wuhlheide
Hirschgarten
Friedrichshagen
Rahnsdorf
Wilhelmshagen
Plänterwald
Baumschulenweg
Oberspree
Betriebsbahnhof Schöneweide
Elsterwerdaer Platz
Biesdorf-Süd
Biesdorf Wuhletal
Kaulsdorf-Nord
Cottbusser Platz
Kaulsdorf Mahlsdorf Birkenstein Hoppegarten
Neuenhagen
Louis-Lewin-Str.
Nöldnerplatz
Samariterstr. Magdalenen-str.
Storkower Str.
Adlershof
Altglienicke
Grünbergallee
Eichwalde
Wildau
Poelchaustr.
Senefelderplatz
Mohrenstr.
Rehberge
Hansaplatz
Bellevue
Bayerischer Platz
Güntzelstr.
Augsburger Str.
Viktoria-Luise-Platz
RathausSchöneberg
Eisenacher Str.
Kleistpark
Möckern-brücke
Friedenau
Hohenzollerndamm
Osdorfer Str.
Schloßstr.
Mahlow
Bernauer Str. Eberswalder Str.
Blaschkoallee
Mühlenbeck-MönchmühleSchönfließBergfelde
Fredersdorf
Petershagen Nord
Strausberg
Hegermühle
Strausberg Stadt
Wilmers-dorfer Str.
Kaiserdamm
Beusselstr.
Friedrichsfelde
Adenauer-platz
Mierendorffplatz
Jakob-Kaiser-Platz
Halensee
Marienfelde
Buckower Chaussee
Johannisthaler Chaussee
Köllnische Heide
Sonnenallee
Frankfurter Tor
Landsberger Allee
Friedrichs-felde Ost
Marzahn
Mendelssohn-Bartholdy-Park
Potsdamer Platz
Branden-burger Tor
Anhalter BhfMoritzplatzKochstr.
Checkpoint Charlie
Seestr.
Kurt-Schumacher-Platz
Karl-Bonhoeffer-Nervenklinik
Pankstr.
Hohenschönhausen
Hellersdorf
Tierpark
Greifswalder Str.
Prenzlauer AlleeSchönhauser Allee
BornholmerStr.
Wollankstr.
Karlshorst
Rosa-Luxemburg-Platz Rosenthaler
PlatzWeinmeisterstr.
Oranien-burger
Str.Naturkunde-
museum
Nordbahnhof
Oranien-burger Tor
Vinetastr.
Turmstr.
Kurfürsten-damm
Schöneberg Alt-Tempelhof
Priesterweg
Lichterfelde WestLichterfelde Ost
Lankwitz
Feuerbachstr.
Wedding
Jannowitz-brücke
Julius-Leber-Brücke
Flughafen Berlin Brandenburg Airport
Waßmannsdorf
Hausvogtei-platz
Stadtmitte
Französische Str.
Fried-richstr.
Nikolassee
Griebnitz-see
Kurfürsten-str.
Kottbusser Tor Görlitzer BhfHallesches TorGleis-dreieckBülow-
str.
Wittenberg-platz
Jungfernheide
KienbergGärten der Welt
Yorckstr.Großgörschenstr.
Yorckstr.
Gesundbrunnen
Zoologischer Garten
Bundestag
FrankfurterAllee
Hohenzollern-platz
Messe ZOB
eE Kein Zugverkehrno service05.06. - 11.12.2018
Ein Verbund.Ein Tarif.
Berlin
CBA
0A
Barrierefrei durch Berlin Step-free access
7 6
ZOB
RB22RE1
RE1 RB22
S-Bahn-/U-Bahn urban rail/underground
Bahn-Regionalverkehrregional railEinige Linien halten nicht überallSome trains do not stop at all stations
Flughafen airport
Fernbahnhof mainline station
Zentraler Omnibusbahnhof bus terminal
VBB-Tarifteilbereiche BerlinVBB fare zones Berlin
Legende Key to symbols
Aufzug lift
Rampe rampnur zur S-Bahn only to urban rail
nur zur U-Bahn only to underground
nur zum Bahn-Regionalverkehronly to regional rail
Stand: 7. Mai 2018© Berliner Verkehrsbetriebe (BVG)600-1-18.1-2
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Berliner Verkehrsbetriebe, bvg.de
July 9, 2018 11 / 48
§6.1 Metro Maps
London, 2018 (Transport for London)
MAYOR OF LONDON
Online maps are strictly for personal use only. To license the Tube map for commercial use please visit tfl.gov.uk/maplicensing
Tube map
Transp
ort fo
r London M
ay 2018
Transport for London May 2018
Key to lines
Metropolitan
Victoria
Circle
Central
Bakerloo
DLR
London Overground
TfL Rail
London Trams
Piccadilly
Waterloo & City
Jubilee
Hammersmith & City
Northern
District
District open weekends and on some public holidays
Emirates Air Linecable car (special fares apply)
Check before you travel§ Brixton No step-free access until September.---------------------------------------------------------------------------§ Heathrow TfL Rail customers should change at Terminals 2 & 3 for free rail transfer to Terminal 5.---------------------------------------------------------------------------§ Hounslow West Step-free access for manual wheelchairs only.---------------------------------------------------------------------------§ Kennington Bank branch trains will not stop between Saturday 26 May and mid-September.---------------------------------------------------------------------------§ Victoria Step-free access is via the Cardinal Place entrance.---------------------------------------------------------------------------§ Services or access at these stations are subject to variation. Please search ‘TfL stations’ for full details.
Transport for London May 2018
Key to symbols Explanation of zones
1
3
4
5
6
2
7
8
9
Station in both zones
Station in both zones
Station in both zones
Station in Zone 9
Station in Zone 6
Station in Zone 5
Station in Zone 3Station in Zone 2
Station in Zone 1
Station in Zone 4
Station in Zone 8
Station in Zone 7
Interchange stations
Step-free access from street to train
Step-free access from street to platform
National Rail
Riverboat services
Airport
Victoria Coach Station
Emirates Air Line cable car
A
B
C
D
E
F
1 2 3 4 5 6 7 8 9
1 2 3 4 5 76 8 9
A
B
C
D
E
F
2 2/3 43
22
2
8 8 6 8
2
4
4
65
9
1
3
2
3
3
3
1
1
33
5
3
5 79 7 7Special fares apply
London Tramsfare zone
5
5
4
4
4
4
46
Special faresapply
River Thames
WestEaling
Regent’s ParkGoodgeStreet
Bayswater
Warren Street
Aldgate
Farringdon
BarbicanRussellSquare
High Street Kensington
Old Street
Green Park
BakerStreet
NottingHill Gate
Mansion House
Temple
OxfordCircus
TowerHill
Westminster
PiccadillyCircus
CharingCross
Holborn
Tower Gateway
Monument
Moorgate
Leicester SquareSt Paul’s
Hyde Park Corner
Knightsbridge
Angel
Queensway
Marble Arch
SouthKensington
SloaneSquare
Covent Garden
LiverpoolStreet
Great PortlandStreet
Bank
Chancery Lane
LancasterGateHolland
Park
Cannon Street
Fenchurch Street
GloucesterRoad St James’s
Park
EustonSquareEdgware
Road
Edgware Road
Embankment
Blackfriars
TottenhamCourt Road
King’s CrossSt Pancras
MarylebonePaddington
Watford High Street
Watford Junction
Bushey
Carpenders Park
Hatch End
North Wembley
South Kenton
Kenton
Wembley Central
Kensal Green
Queen’s Park
Stonebridge Park
Bethnal Green
Cambridge Heath
London Fields
Harlesden
Willesden Junction
Headstone Lane
Harrow &Wealdstone
Kilburn Park
WarwickAvenue
Maida ValeEuston
NewCross Gate
Imperial Wharf
ClaphamJunction
Crystal Palace Norwood Junction
Sydenham
Forest Hill
Anerley
Penge West
Honor Oak Park
Brockley
Wapping
New Cross
Queens RoadPeckham
Peckham Rye
Denmark Hill
Surrey Quays
Whitechapel
WandsworthRoad
Rotherhithe
ShoreditchHigh Street
Haggerston
Hoxton
Shepherd’sBush
Shadwell
CanadaWater
Fulham Broadway
West Brompton
Parsons Green
Putney Bridge
East Putney
Southfields
Wimbledon Park
Wimbledon
Kensington(Olympia)
AldgateEast
Bethnal Green Mile End
Dalston Kingsland
HackneyWick
Homerton
HackneyCentral
RectoryRoad
HackneyDowns
Theydon Bois
Epping
Debden
Loughton
Buckhurst Hill
Leytonstone
Wood StreetBruce Grove
White Hart Lane
Silver Street
Edmonton Green
Southbury
Turkey Street
Theobalds Grove
Cheshunt
Enfield Town
StamfordHill
Bush HillPark
Highams Park
Chingford
Leyton
Woodford
South Woodford
Snaresbrook
Hainault
Fairlop
Barkingside
Newbury Park
Stratford
RodingValley
GrangeHill
Chigwell
Redbridge
GantsHill
Wanstead
Dalston Junction
Canonbury
Stepney Green
SevenSisters
Highbury &Islington
TottenhamHale
Clapton
St James Street
StokeNewington
Dagenham East
Dagenham Heathway
Becontree
Upney
Upminster
Upminster Bridge
Hornchurch
Elm Park
Ilford
Goodmayes
Chadwell Heath
Romford
Gidea Park
Harold Wood
Shenfield
Brentwood
Seven Kings
HarringayGreenLanes
LeytonstoneHigh Road
LeytonMidland Road
Emerson ParkSouth Tottenham
Barking
East Ham
Plaistow
Upton Park
Upper Holloway
CrouchHill
Gospel Oak
BowChurch
WestHam
BowRoad
Bromley-by-Bow
Island Gardens
Greenwich
Deptford Bridge
South Quay
Crossharbour
Mudchute
Heron Quays
West IndiaQuay
Elverson Road
Devons Road
Langdon Park
All Saints
Canary Wharf
Cutty Sark for Maritime Greenwich
Lewisham
West Silvertown
EmiratesRoyal Docks
EmiratesGreenwichPeninsula
PontoonDock
LondonCity Airport
WoolwichArsenal
King George V
Custom House for ExCeL
Prince Regent
Royal Albert
Beckton Park
Cyprus
Beckton
Gallions Reach
Westferry Blackwall
RoyalVictoria
CanningTown
PoplarLimehouse
EastIndia
StratfordInternational
Star Lane
NorthGreenwich
Maryland
Manor Park
Oakwood
Cockfosters
Southgate
Arnos Grove
Bounds Green
Turnpike Lane
Wood Green
Manor House
FinsburyPark
Arsenal
KentishTown West
Holloway Road
Caledonian Road
Mill Hill East
Edgware
Burnt Oak
Colindale
Hendon Central
Brent Cross
Golders Green
Hampstead
Belsize Park
Chalk Farm
Camden Town
High Barnet
Totteridge & Whetstone
Woodside Park
West Finchley
Finchley Central
East Finchley
Highgate
Archway
Tufnell Park
Kentish Town
MorningtonCrescent
CamdenRoad
CaledonianRoad &
Barnsbury
Amersham
Chorleywood
Rickmansworth
Chalfont &Latimer
Chesham
Moor Park
Croxley
Watford
Northwood
Northwood Hills
Pinner
North Harrow
Harrow-on-the-Hill
NorthwickPark
PrestonRoad
Wembley Park
Rayners Lane
Stanmore
Canons Park
Queensbury
Kingsbury
Neasden
Dollis Hill
Willesden Green
Swiss Cottage
KilburnWestHampstead
Finchley Road
WestHarrow
IckenhamUxbridge
Hillingdon RuislipRuislip Manor
Eastcote
St John’s Wood
HeathrowTerminal 5
Northfields
Boston Manor
SouthEaling
Osterley
Hounslow Central
Hounslow East
Hounslow West
Hatton Cross
Perivale
Hanger Lane
RuislipGardens
South Ruislip
Greenford
Northolt
South Harrow
Sudbury Hill
Sudbury Town
Alperton
Park Royal
North Ealing
West Ruislip
Ealing Common
Gunnersbury
Kew Gardens
Richmond
Acton Town
ChiswickPark
TurnhamGreen
StamfordBrook
RavenscourtPark
WestKensington
BaronsCourt
Earl’sCourt
Shepherd’sBush Market
Goldhawk Road
Hammersmith
Wood Lane
WhiteCity
Finchley Road& Frognal
KensalRise
BrondesburyPark
Brondesbury
KilburnHigh Road
SouthHampstead
WestActon
NorthActon
EastActon
Southwark
Waterloo
London Bridge Bermondsey
Vauxhall
Lambeth NorthPimlico
Stockwell
Brixton
Elephant & Castle
Oval
Kennington
Borough
Clapham North
Clapham High Street
Clapham Common
Clapham South
Balham
Tooting Bec
Tooting Broadway
Colliers Wood
South Wimbledon
Morden
Latimer Road
Ladbroke Grove
Royal Oak
Westbourne Park
PuddingMill Lane
Acton Central
South Acton
HampsteadHeath
Stratford High Street
Abbey Road
WoodgrangePark
WalthamstowQueen’s Road
West Croydon
BeckenhamJunction
Elmers End
Harrington Road
Arena
DundonaldRoad
Merton Park Woodside
Blackhorse Lane
Addiscombe
AvenueRoad
Sandilands
WellesleyRoad
Reeves Corner
Mitcham BeddingtonLane
AmpereWay
WandlePark
Centrale
ChurchStreet
BelgraveWalk
PhippsBridge
MordenRoad
TherapiaLane
WaddonMarsh
GeorgeStreet
LebanonRoad
EastCroydon
BeckenhamRoad
MitchamJunction
Birkbeck
WalthamstowCentral
WansteadPark
Blackhorse Road
Forest Gate
Victoria
HeathrowTerminal 4
HeathrowTerminals 2 & 3
ActonMain Line
EalingBroadway
BondStreet
Southall
Hanwell
Hayes &Harlington
Coombe Lane
Lloyd Park King Henry’sDriveFieldway
NewAddington
GravelHill
AddingtonVillage
Transport for London, tfl.gov.uk
July 9, 2018 12 / 48
§6.1 Metro Maps
Saint Petersburg, 2018
Петербургский Метрополитен, metro.spb.ru
July 9, 2018 13 / 48
Chapter 6
Metro Map Drawing
§6.2 Planar Graphs
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§6.2 Planar Graphs
Planar EmbeddingsLet G = (V ,E ) be a (simple) graph.
DefinitionA planar embedding of G consists of an injective map ψ : V → R2, and afamily of continuous and injective functions fe : [0, 1]→ R2 for each edgee ∈ E such that
I fvw (0) = ψ(v) and fvw (1) = ψ(w) for all vw ∈ E ,
I fe((0, 1)) ∩⋃
e′ 6=e fe′([0, 1]) = ∅ for all e ∈ E .
G is called planar if it admits some planar embedding.
RemarkThis embeds a graph into the plane using simple Jordan curves.
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§6.2 Planar Graphs
Faces
DefinitionA connected component of R2 \
⋃e∈E fe([0, 1]) is called a face.
ObservationsI There is exactly one unbounded face (Jordan curve theorem).
I The boundary of a bounded face F gives rise to a face circuit CF in G .
Lemma{CF | F is a bounded face} is an undirected cycle basis of G .
Proof.Let C be a circuit in G . Then R2 \
⋃e∈E(C) fe([0, 1]) is the union of a
bounded and an unbounded component. Let F1, . . . ,Fk be the facescontained in the bounded component. Then C = CF1 + · · ·+ CFk
.Suppose
∑F bounded face λFCF = 0 for some λF ∈ F2. If e is an edge
between a bounded face F and the outer face, then λF = 0. Proceed byinduction on the number of bounded faces.
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§6.2 Planar Graphs
Euler’s formulaLet G be a planar graph with n vertices, m edges, f faces and c connectedcomponents.
Corollary (Euler, 1758)
f − 1 = m − n + c .
LemmaSuppose that G is connected and n ≥ 3.
(1) 2m ≥ 3f (2) m ≤ 3n − 6 (3) f ≤ 2n − 4
Proof.
(1) Handshake: 2m =∑
v∈V deg(v) =∑
F face #{vertices along F} ≥ 3f .If n ≥ 3, then the outer face contains at least 3 vertices.
(2) 2m ≥ 3f = 3(m − n + 2) = 3m − 3n + 6 ⇒ m ≤ 3n − 6.
(3) 3n − 6 ≥ m = n + f − 2 ⇒ f ≤ 2n − 4.
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§6.2 Planar Graphs
K5 and K3,3
LemmaThe complete graph K5 is not planar.
Proof.We have n = 5 and m =
(52
)= 10, so m = 10 > 9 = 3 · 5− 6 = 3n− 6.
LemmaThe complete bipartite graph K3,3 is not planar.
Proof.Every circuit in K3,3 has length at least 4. In particular, if K3,3 was planar,then 2m ≥ 4(f − 1) + 3 = 4(m − n + 1) + 3 = 4m − 4n + 7 and therefore2m ≤ 4n − 7. But 2m = 2 · 32 = 18 and 4n − 7 = 4 · 6− 7 = 17.
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§6.2 Planar Graphs
Minors
DefinitionLet G and M be undirected graphs. M is a minor of G if a series of thefollowing operations transforms G into M:
I delete a vertex
I delete an edge
I contract an edge
Theorem (Wagner, 1937)
An undirected graph is planar if and only if it contains neither K5 nor K3,3
as a minor.
Proof.(⇒) Any minor of a planar graph must be planar.(⇐) Omitted.
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§6.2 Planar Graphs
More on Minors
Theorem (Robertson/Seymour, 2004)
A family of graphs is closed under taking minors if and only if it has afinite number of minimal forbidden minors.
Example
Planar graphs: K5, K3,3 Forests: C3 (circuit on 3 vertices)
Theorem (Robertson/Seymour, 1995)
For any fixed minor M, there is a O(n3) algorithm deciding whether agraph on n vertices has M as a minor.
This is nice, but the proof is not constructive. For planarity testing, thereare better (and explicit) algorithms:
Theorem (Hopcroft/Tarjan, 1974)
Planarity testing can be done in O(n) time.
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§6.2 Planar Graphs
Subdivisions
DefinitionLet G and M be undirected graphs. M is a subdivision of G if it isobtained from G by consecutively replacing an edge uw with two edgesuv , vw , inserting a new vertex v .
u w→
u v w
The reverse process is called smoothing.
Theorem (Kuratowski, 1930)
An undirected graph is planar if and only if it contains no subgraph that isa subdivision of K5 or K3,3.
Proof.(⇒) Any subgraph and any smoothing of a planar graph must be planar.(⇐) Omitted.
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§6.2 Planar Graphs
2-basesObservationLet G be a planar embedded graph. Then every edge is contained in atmost two face circuits.
DefinitionA cycle basis B of a graph is called a 2-basis if every edge is contained inat most two cycles of B.
Theorem (MacLane, 1937)A graph is planar if and only if it admits a 2-basis.
Proof (O’Neill, 1973).Planar graphs have 2-bases. The following would prove the converse:
(1) Admitting a 2-basis is closed under taking subgraphs and smoothings.
(2) K5 and K3,3 do not admit a 2-basis.
Then any graph with a 2-basis does not contain a subdivision of K5 orK3,3, and must hence be planar by Kuratowski’s theorem.July 9, 2018 22 / 48
§6.2 Planar Graphs
MacLane’s planarity criterion
Proof of (1).
Let {C1, . . . ,Cµ} be a 2-basis of a graph G = (V ,E ).
Deleting an edge e ∈ E : If e is not contained in any cycle, nothinghappens. Otherwise this reduces the cyclomatic number by 1. If e iscontained in a single cycle (w.l.o.g. C1), then {C2, . . . ,Cµ} is a 2-basis of(V ,E \ {e}). If e is contained in two cycles (w.l.o.g. C1,C2), then{C1 + C2, . . . ,Cµ} is a 2-basis of (V ,E \ {e}).
Deleting a vertex: Delete first all adjacent edges. Removing an isolatedvertex does not affect the cyclomatic number.
Smoothing uv and vw to uw : The columns of uv and vw in the cyclematrix are the same, so smoothing does not change anything about the2-basis.
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§6.2 Planar Graphs
MacLane’s planarity criterion
Proof of (2).
Let {C1, . . . ,C6} be a 2-basis for K5. Set C7 :=∑6
i=1 Ci . Observe thatC7 6= 0 and
∑7i=1 Ci ,e = 0 ∈ F2 for all e ∈ E . In particular, every edge is
contained in exactly two of the cycles C1, . . . ,C7. Hence∑7i=1 |E (Ci )| = 2|E (K5)| = 20.
On the other hand, |E (Ci )| ≥ 3 for all i , so∑7
i=1 |E (Ci )| ≥ 21.
Now let {C1, . . . ,C4} be a 2-basis for K3,3. Set C5 :=∑4
i=1 Ci . Again∑5i=1 Ci ,e = 0 for all e ∈ E , so
∑5i=1 |E (Ci )| = 2|E (K3,3)| = 18. Since
|E (Ci )| ≥ 4 for all i ,∑5
i=1 |E (Ci )| ≥ 20.
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§6.2 Planar Graphs
More on 2-bases
LemmaThe cycle matrix of a 2-basis of a directed graph is totally unimodular. Inparticular, any 2-basis is integral.
Proof.Orient all cycles counter-clockwise w.r.t. some planar embedding. Proceedby induction on the size q of a quadratic submatrix A of the cycle matrix.The case q = 1 is clear. If q ≥ 2, there are two cases:
(1) A contains a column with a single non-zero entry. Use Laplaceexpansion and induction.
(2) All columns of A have at least two non-zero entries. Since we have a2-basis, there are exactly two non-zeros, a +1 and a −1. So all rows ofA add up to 0, showing detA = 0.
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§6.2 Planar Graphs
Graph drawings
Let G be a planar graph.
Types of planar embeddings
I polygonal : All edges are embedded as polygonal arcs.
I straight line: All edges are drawn as straight line segments.
I rectilinear : All edges are drawn as straight line segments with slopesk · 90◦, k ∈ Z.
I octilinear : All edges are drawn as straight line segments with slopesk · 45◦, k ∈ Z.
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§6.2 Planar Graphs
Combinatorial type
DefinitionLet G = (V ,E ) be a planar graph with some planar embedding. For eachvertex v ∈ V , the embedding defines an ordering of the neighborhood of vby counter-clockwise sorting of the edges incident to v . This is thecombinatorial type of the embedding.
A
B
CD
A: (B, D, C)
A
B C
D
A: (D, B, C)
This defines an equivalence relation on planar embeddings of G .
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§6.2 Planar Graphs
Straight line drawings
Theorem (Fary, 1948)
Any simple planar graph has a straight line drawing.
Proof.Let G be a simple connected planar graph with n ≥ 3 vertices, embeddedinto the plane. W.l.o.g. G is maximally planar, i.e., m = 3n − 6, and everybounded face is a triangle.
Claim: If uvw is a triangle, then there is a straight line embedding of thesame combinatorial type where u, v ,w are the vertices along the outer face.
This is easy for n = 3. Thus let n ≥ 4, and let uvw be a triangle. Not allof u, v ,w have degree 2 (connectedness). Assume that the other n − 3vertices have degree at least 6. Then∑
v∈Vdeg(v) ≥ 6(n − 3) + 7 = 6n − 11 > 6n − 12 = 2m,
contradicting the Handshaking Lemma.
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§6.2 Planar Graphs
Straight line drawings
Proof (cont.)
In particular, we find a vertex z different from u, v ,w with deg(z) ≤ 5.Now remove z from the graph and retriangulate the new face F . The newgraph has n − 1 vertices and – by induction – admits a straight lineembedding of the same combinatorial type where u, v ,w are the verticesalong the outer face. In particular, the face F has become a simplepolygon with at most 5 sides.
By the art gallery theorem (with b5/3c = 1 guards), there is a point insidethis polygon that can be connected to all vertices of F by non-crossingstraight lines (Short proof: Triangulations of simple polygons admit a3-coloring.)
Theorem (De Fraysseix/Pach/Pollack, 1990)
Straight line drawings on a grid can be found in linear time.
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§6.2 Planar Graphs
Straight line drawings: Example
z
delete z−−−−→in
du
ction
−−−−−→
insert z←−−−−
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§6.2 Planar Graphs
Rectilinear and octilinear drawingsLet G be a graph.
Theorem (Garg/Tamassia, 2001)
It is NP-hard to decide whether G admits a rectilinear drawing.
Theorem (Tamassia, 1987)
Fix some planar embedding of G . There is a polynomial-time algorithmthat decides whether G admits a rectilinear drawing preserving thecombinatorial type.
Theorem (Nollenburg, 2005)
Fix some planar embedding of G . It is NP-hard to decide whether Gadmits an octilinear drawing preserving the combinatorial type.
Both NP-hardness proofs reduce (variants of) the 3-SAT problem.
RemarkClearly, if G admits a rectilinear (octilinear) drawing, then deg(v) ≤ 4(≤ 8) for all v ∈ V (G ).July 9, 2018 31 / 48
Chapter 6
Metro Map Drawing
§6.3 Octilinear Layout Computation
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§6.3 Octilinear Layout Computation
Requirements
Design Principles (taken from Nollenburg, 2011)
I Preserve the combinatorial type.
I Octilinear drawing: All edges are drawn as line segments with slopesk · 45◦ for k ∈ {0, 1, . . . , 7}.
I Lines should avoid sharp bends, and pass straight throughinterchanges.
I Ensure a minimum distance between stations, and stations andnon-incident edges.
I Minimize geometric distortion.
I Use uniform edge lengths.
I Use large angular resolution.
I Place station labels in a readable way.
I . . .
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§6.3 Octilinear Layout Computation
Metro Map Layout Problem
Input
I a line network consisting of a graph G = (V ,E ) of maximum degree 8and a line cover L
I a planar embedding of G , e.g., by geographical coordinates
Output
I a map ψ : V → R2 inducing an octilinear drawing of G
I satisfying/optimizing design principles
Solution Methods
I metaheuristics (hill climbing, simulated annealing, ant colonies, . . . )
I local optimization: least squares
I global optimization: mixed integer programming
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§6.3 Octilinear Layout Computation
U-Bahn Berlin: geographical layout
173 vertices, 184 edges, 10 lines
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§6.3 Octilinear Layout Computation
U-Bahn Berlin: curvilinear layout
least squares method (Wang/Chi 2011, Wang/Peng 2016)
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§6.3 Octilinear Layout Computation
U-Bahn Berlin: octilinear layout?
least squares method (Wang/Chi 2011, Wang/Peng 2016), naive python/cvxopt implementation: 36 s
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§6.3 Octilinear Layout Computation
Octilinear Layout MIPWe will use the formulation due to Nollenburg (2005):
Hard constraints (feasibility)
I octilinearity
I combinatorial type preservation
I minimum edge length
I minimum distance for non-adjacent edges
In particular, a feasible solution guarantees octilinearity.
Soft constraints (objective)
I bend minimization
I preservation of relative positions for adjacent stations
I minimum total edge length
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§6.3 Octilinear Layout Computation
Octilinear Layout MIP: Basic variables
I vertex coordinates xv , yv ∈ [0, |V |] for v ∈ V
I additional vertex coordinates zv := xv + yv , wv := xv − yvI edge directions dirvw , dirwv ∈ {0, 1, . . . , 7} for {v ,w} ∈ E
I original directions secvw :=[⌊
^(v ,w)45◦ + 1
2
⌋]8,
where ^(v ,w) ∈ (−180◦, 180◦] is the slope of the edge {v ,w}
Nollenburg (2005)
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§6.3 Octilinear Layout Computation
Octilinear Layout MIP: DirectionsI binary variables αi ,vw for i ∈ {−1, 0, 1} and {v ,w} ∈ E such thatαi ,vw = 1⇔ edge {v ,w} is in original direction +i · 45◦
α−1,vw + α0,vw + α1,vw = 1, {v ,w} ∈ E
dirvw + Mαi ,vw ≤ M + [secvw + i ]8, i ∈ {−1, 0, 1}, {v ,w} ∈ E
dirvw −Mαi ,vw ≥ −M + [secvw + i ]8, i ∈ {−1, 0, 1}, {v ,w} ∈ E
dirwv + Mαi ,vw ≤ M + [secvw + i + 4]8, i ∈ {−1, 0, 1}, {v ,w} ∈ E
dirwv −Mαi ,vw ≥ −M + [secvw + i + 4]8, i ∈ {−1, 0, 1}, {v ,w} ∈ E
M � 0 is a large constantI relating directions with coordinates, e.g., for secvw = 2 and α0,vw :
xv − xw + Mα0,vw ≤ M
xv − xw −Mα0,vw ≥ −Myv − yw + Mα0,vw ≤ M −minimum edge length
If α0,vw = 1, then xv = xw and yw ≥ yv + minimum edge length.July 9, 2018 40 / 48
§6.3 Octilinear Layout Computation
Octilinear Layout MIP: Planar embedding
I preservation of combinatorial type: binary variables βi ,v fori ∈ {1, . . . , deg(v)} and v ∈ V ,
deg(v)∑i=1
βi ,v = 1, v ∈ V ,
dirv ,wj − dirv ,w[j−1]deg(v)+ Mβj(v) ≥ 1, v ∈ V , j = 1, . . . , deg(v),
where w1, . . . ,wdeg(v) are the neighbors of v in counter-clockwiseorder
I planarity constraints: modeled by binary variables γi ,e,e′ fori ∈ {0, . . . , 7} and e, e ′ ∈ E non-incident (large number!)
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§6.3 Octilinear Layout Computation
Octilinear Layout MIP: ObjectiveI total edge length objective: by new variables lenvw , vw ∈ E , e.g., for
secvw = 2 and α0,vw :
yv − yw + Mα0,vw = M − lenvw
If α0,vw = 1, then xv = xw , and yw ≥ yv = lenvw . Edge lengths aremeasured w.r.t. | · |∞: a diagonal line segment (x , y)→ (x + 1, y + 1)has length 1.
I bend objective: for each three consecutive stations (u, v ,w) on a lineof the network, add the constraints
benduvw =
{|diruv − dirvw | if |diruv − dirvw | ≤ 4
8− |diruv − dirvw | if |diruv − dirvw | ≥ 5
I relative position objective:
−M · relvw ≤ dirvw − secvw ≤ M · relvw , vw ∈ E
I objective function: λ1∑
vw lenvw + λ2∑
uvw benduvw + λ3∑
vw relvw
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§6.3 Octilinear Layout Computation
Octilinear Layout MIP: Practical Aspects
SizeSuppose that G has n vertices and m edges. Let m′ :=
∑`∈L |E (`)|. Then
the MIP formulation uses uses O(n + m′ + m2) variables and constraints.
Example
The Berlin U-Bahn example needs
I 3 046 variables (1 623 binary, 546 integer, 877 continuous) and 7 149constraints without planarity constraints.
I 137 158 variables (135 735 binary, 546 integer, 877 continuous) and560 361 constraints with planarity constraints
Pre-processingI Do not use all planarity constraints (heuristics, lazy constraints).
I Contract vertices of degree 2.
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§6.3 Octilinear Layout Computation
Crossing minimizationConsider a line network (G ,L). The metro line crossing minimizationproblem is to find an ordering of the lines at each station such that thetotal number of crossings of lines is minimized.
Results
I The problem is in general NP-hard (Fink/Pupyrev, 2013).
I There is an integer programming formulation (Asquith et. al., 2008).
I The key step is to determine the ordering of the lines at theirterminals.
I For a fixed ordering at the terminals, there is a polynomial-timealgorithm.
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§6.3 Octilinear Layout Computation
U-Bahn Berlin: octilinear layout!
mixed integer programming method (Nollenburg 2005), solution found by CPLEX 12.7.1 after 66 s, optimality: 15 min 55 s
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§6.3 Octilinear Layout Computation
S-Bahn Berlin: geographical layout
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§6.3 Octilinear Layout Computation
S-Bahn Berlin: curvilinear layout
least squares method, naive python/cvxopt implementation: 31 s
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§6.3 Octilinear Layout Computation
S-Bahn Berlin: octilinear layout
mixed integer programming method, solution found by CPLEX 12.7.1 after 25 s, optimality gap: ≤ 3%
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