Mathematical Aspects of Public Transportation Networks · commons.wikimedia.org July 9, 2018 7 / 48. x6.1 Metro Maps London, 1933 (Harry Beck) July 9, 2018 8 / 48. ... S 4 1 >

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)


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)


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

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RB14

RE3 RE7

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RE1

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RE2RB24

RE6. RB20

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RE4.RB13

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RE6 RB55

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RB22 RB36

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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 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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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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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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U-Bahn Berlin: geographical layout

173 vertices, 184 edges, 10 lines

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U-Bahn Berlin: curvilinear layout

least squares method (Wang/Chi 2011, Wang/Peng 2016)

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U-Bahn Berlin: octilinear layout?

least squares method (Wang/Chi 2011, Wang/Peng 2016), naive python/cvxopt implementation: 36 s

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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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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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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


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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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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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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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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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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S-Bahn Berlin: geographical layout

July 9, 2018 46 / 48


S-Bahn Berlin: curvilinear layout

least squares method, naive python/cvxopt implementation: 31 s

July 9, 2018 47 / 48


S-Bahn Berlin: octilinear layout

mixed integer programming method, solution found by CPLEX 12.7.1 after 25 s, optimality gap: ≤ 3%

July 9, 2018 48 / 48

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