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On the number of touching pairs in a set of planar curves

Citation for published version (APA):

Györgyi , P., Hujter , B., & Kisfaludi-Bak, S. (2018). On the number of touching pairs in a set of planar curves.

Computational Geometry, 67(Januari 2018), 29-37 . https://doi.org/10.1016/j.comgeo.2017.10.004

DOI:

10.1016/j.comgeo.2017.10.004

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Published: 01/01/2018

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Contents lists available atScienceDirect

Computational

Geometry:

Theory

and

Applications

www.elsevier.com/locate/comgeo

On

the

number

of

touching

pairs

in

a

set

of

planar

curves

Péter Györgyi

a

,

Bálint Hujter

b

,

Sándor Kisfaludi-Bak

c,

aInstituteforComputerScienceandControl,Budapest,Hungary

bMTA-ELTEEgerváryResearchGroup,EötvösUniversity,Budapest,Hungary cDepartmentofMathematicsandComputerScience,TUEindhoven,TheNetherlands

a

r

t

i

c

l

e

i

n

f

o

a

b

s

t

r

a

c

t

Articlehistory:

Received27October2015

Receivedinrevisedform15February2017 Accepted15September2017

Keywords:

Combinatorialgeometry Touchingcurves Pseudo-segments

Given aset ofplanar curves (Jordan arcs), eachpair of whichmeets — either crosses or touches — exactlyonce, we establish an upper boundon the number of touchings. We showthatsuchacurvefamilyhas O(t2n)touchings,wheret isthenumberoffaces

inthe curvearrangementthat containsatleastone endpointofoneofthe curves.Our methodreliesonfindingspecialsubsetsofcurvescalledquasi-gridsincurvefamilies;this givessomestructuralinsightintocurvefamilieswithahighnumberoftouchings.

©2017ElsevierB.V.Allrightsreserved.

1. Introduction

Thecombinatorialexaminationofincidencesintheplanehasproventobeafruitfulareaofresearch.Thefirstseminal re-sultsarethecrossinglemmathatestablishesalowerboundonthenumberofedgecrossingsinaplanardrawingofagraph (Ajtaietal.,Leighton[1,2]), andthetheorembySzemerédiandTrotter[3],concerningthenumberofincidencesbetween linesandpoints.Soon,theincidencesofmoregeneralgeometricobjects(segments,circles,algebraiccurves,pseudo-circles, Jordanarcs,etc.)becamethecenterofattention[4–9].Withtheadditionofcurves,thedistinctionbetweentouchingsand crossingsisinorder.

Usually,the curvesareeither Jordanarcs,i.e., theimage ofan injectivecontinuous function

ϕ

:

[0

,

1]

→ R

2,orclosed Jordancurves,where

ϕ

isinjectiveon[0

,

1

)

and

ϕ

(

0

)

=

ϕ

(

1

)

.Generally,itissupposedthat thecurvesintersectinafinite numberofpoints,andthatthecurvesareingeneralposition:threecurvescannotmeetatonepoint,and(incaseofJordan arcs) an endpoint ofa curve doesnot lie onanyother curve. (Fortechnical purposes,we willallow curve endpoints to coincideinsomeproofs.)

LetP beapointwherecurvea andb meet.Takeacircle

γ

withcenterP andasmallenoughradiussothatitintersects botha andb twice,andthediskdeterminedby

γ

isdisjointfromalltheothercurves,andcontainsnootherintersections ofa andb.Labeltheintersectionpoints of

γ

andthetwocurveswiththenameofthecurve.Wesaythat a andb cross

in P ifthecyclicalpermutationoflabelsaround

γ

isabab,anda andb touch in P ifthecyclical permutationoflabelsis

aabb.Inafamilyofcurves,let X bethesetofcrossingsandT bethesetoftouchings.

TheRichter–Thomassen conjecture[10]statesthatgivenacollectionofn pairwiseintersectingclosedJordancurvesin generalpositionintheplane,thenumberofcrossingsisatleast

(

1

o

(

1

))

n2.AproofoftheRichter–Thomassenconjecture

hasrecentlybeenpublishedbyPachetal.[11].TheyshowthatthesameresultholdsforJordanarcsaswell.

*

Correspondingauthor.

E-mailaddresses:gyorgyi.peter@sztaki.mta.hu(P. Györgyi),hujterb@cs.elte.hu(B. Hujter),s.kisfaludi.bak@tue.nl(S. Kisfaludi-Bak). https://doi.org/10.1016/j.comgeo.2017.10.004

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30 P. Györgyi et al. / Computational Geometry 67 (2018) 29–37

It would be preferableto get moreaccurate bounds forthe ratiooftouchings andcrossings. Fox etal. constructed a family of x-monotone curves withratio

|

X

|/|

T

|

=

O

(

log n

)

[12]. Ifwe restrict the number of intersections betweenany two curves,then it isconjectured that the above ratio is much higher. It has been shownthat a familyof intersecting pseudo-circles (i.e., a set of closed Jordan-curves, any two of which intersect exactly once or twice) has at most O

(

n

)

touchings[7].WewouldliketoexamineasimilarstatementforJordanarcs.

A familyofJordanarcs inwhichanypairofcurvesintersect atmostonce (apartfromtheendpoints)willbe calleda

familyofpseudo-segments.OurstartingpointisthisconjectureofJánosPach[13]:

Conjecture1.Let

C

beafamilyofpseudo-segments.Supposethatanypairofcurvesin

C

intersectexactlyonce.Thenthenumberof touchingsin

C

isO

(

n

)

.

Afamilyofpseudo-segmentsisintersecting ifeverypairofcurvesintersects(i.e.,eithertouchesorcrosses)exactlyonce outsidetheirendpoints.

Two importantspecial cases ofthe above are the cases ofgrounded and double-grounded curves.(The definitions are takenverbatimfrom[9].)Acollection

C

ofcurvesisgrounded ifthereis aclosedJordancurve g calledground suchthat eachcurvein

C

hasoneendpointon g andtherestofthecurveisintheexteriorofg.Thecollectionisdoublegrounded if

therearedisjointclosedJordancurvesg1 andg2suchthateachcurvec

C

hasoneendpointong1 andtheotherendpoint

on g2,andtherestofc isdisjointfromboth g1 andg2.

According to our knowledge the best upper bound is O

(

nlog n

)

for the number of touchings in a double-grounded

x-monotonefamilyofpseudo-segments[14]andwedonotknowany(non-trivial)resultforthegroundedcase.

1.1. Ourcontribution

Let

C

beanintersectingfamilyofpseudo-segments.Thereisaplanargraphdrawingthatcorrespondstothisfamily:the vertices arethe crossingsandtouchings,andtheedges are thesectionsofthe curvesbetweenneighboringintersections. (Notice that the sections betweencurve endpoints andthe neighboring intersections are not represented inthis graph.) Consider the faces ofthisplanar graph drawing.LettC bethe numberof facesthat contain an endpointof atleastone curvein

C

.Ourmaintheoremcanbestatedasfollows:

Theorem2.Let

C

beann-elementintersectingfamilyofpseudo-segmentsontheEuclideanplane.Thenthenumberoftouchings betweenthecurvesis f

(

n

)

=

O

(

t2Cn

)

.

If tC is constant, this theorem settles Conjecture 1. Note that this includes the case when

C

is a double-grounded intersectingfamilyofpseudo-segments:

Corollary3.Let

C

beann-elementdouble-groundedintersectingfamilyofpseudo-segments.Thenthenumberoftouchingsbetween thecurvesisO

(

n

)

.

Acarefullookattheproofofthemaintheoremyieldsthefollowingresultforgroundedintersectingfamiliesof pseudo-segments:

Theorem4.Let

C

beann-elementgroundedintersectingfamilyofpseudo-segments.Thenthenumberoftouchingsbetweenthecurves isO

(

tCn

)

.

Theintuitionbehind ourapproachcanbedescribedasfollows.Curvesstartinginthesamefaceofanarrangementcan bethoughtofascurveshavingthesameendpoints.Acurvegoingfrompoint A to B thattouchessomeothercurve g can

dothattouchingonlyinaconstantnumberofways,dependingonwhichside ofg istouchedandinwhichdirection.We observethat acollectionofcurvesgoing from A to B must thereforecontainasubcollection thattouch g the sameway, andthesecurvesmusthaveaveryspecialgrid-likestructure,whichwecallquasi-grids.

Itturnsoutthatquasi-gridsalways emergewhenwetaketwogridfamiliesofpseudo-segments,onecontaining curves from A to B,the other containingcurvesfrom C toD. Notethat acurve touching allcurvesina largequasi-grid hasto lie outsidethe“gridcells”, sinceitcannotcrossthequasi-gridcurves,andwithina“gridcell”itcould onlyreachatmost fourcurves.Ifwefindtwocurvestouchingthesamelargequasi-grid,then(intuitively)thosetwocurveswouldhavemany intersections —thisisnotpossiblein anintersecting familyofpseudo-segments.Weshow that thenumberoftouchings betweenapairoffixedendpointcurvefamilies islinearinthesizeofthesefamilies. Wethenusethisobservationtoget theboundonthetotalnumberoftouchings.

2. Proofofthemaintheorem

The rigorous proof of ourmain theorem isbased upon a key lemma. Its proof anticipates anduses several technical lemmaswhicharedetailedinSections3and4.

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Beforestatingthekey lemma,we introducesomenotations.Thenotation g



h meansthatcurves g andh toucheach other.If A and B are(notnecessarilydistinct)points ontheplane,then

C(

A

,

B

)

denotes thesetofdirectedcurvesgoing from A to B.Note thatherewe considercurves asdirected onesfortechnicalreasons (forexample,we canrefer tothe sidesofadirectedcurveasleft andright).

Lemma5.LetA

,

B

,

C

,

D benotnecessarilydistinctpointsontheplane,and

C

1and

C

2befinitedisjointcurvefamiliesfrom

C(

A

,

B

)

and

C(

C

,

D

)

,respectively.If

C

1

C

2isanintersectingfamilyofpseudo-segments,then

1. thenumberofc1



c2touchingswherec1

C

1andc2

C

2isO

(

|

C

1

C

2

|)

;

2. thenumberoftouchingsbetweencurvesof

C

iisO

(

|

C

i

|)

(

i

=

1

,

2

)

.

Proof. Weonlyconsiderthefirstclaim,thesecondcanbeprovenwiththesametools.Supposeforcontradictionthatthere are

ω

(

|

C

1

C

2

|)

instancesofc1



c2 touchings.

Let K bealarge constant.Without lossofgenerality, wecan supposethat eachcurve of

C

i touchesatleast K curves

of

C

j.Toseethis,considerfirstthebipartite graphG withvertexset

C

1

C

2,wheretheedges correspondtothec1



c2

touchings(c1

C

1 andc2

C

2).IfG hasverticesofdegreelessthan K ,thendeletethoseverticesandtheincidentedges.

IteratethisprocedureuntiltheminimumdegreeisatleastK orthegraphisempty.IfG hadatleastK

|

C

1

C

2

|

edges,then

thisprocedurecannotresultinanemptygraph.

Let g

C

1 be an arbitrarycurve. ByLemma 10, there is a quasi-grid withrespect to g formed by atleast K

/

48

>

3

curves.Aquasi-gridisdepictedinFig. 1,theprecisedefinitionisgiveninDefinition 6.

Consider an “inner” curve h in this quasi-grid. By Lemma 11, if a curve touches h, then it must also touch g or a neighboringcurveofh inthequasi-grid.Byourstartingassumption,atleastK curvestouchh.ThenbyLemma 10,atleast

K

/

48 ofthecurvestouchingh mustalsotouchanotherspecificcurveh,andatleastK

/(

48

)

2 oftheseformaquasi-grid

Q

withrespecttobothh andh.

Therefore,bychoosingK

4

·

482

+

1,thequasi-grid

Q

canbeforcedtocontainatleastfivecurves.Thisisa contradic-tionbyLemma 13.

2

NextweshowhowLemma 5impliesTheorem 2.Lett

=

tC.

Proof ofTheorem 2. Consider theplanargraphdrawingthat correspondsto

C

.Letthe facesofthisplanargraphdrawing thatcontainanendpointofatleastonecurvein

C

be:F1

,

F2

,

. . . ,

Ft.

Fori

=

1

,

2

,

. . . ,

t,let Pi beanarbitrarypointintheinteriorof Fi notincidentto anycurvein

C

.Eachcurve endpoint

inside Ficanbeconnectedto Piwithoutaddinganyintersectionsbetweenthecurvesof

C

withtheexceptionofPi.Let

C



bethefamilyofpseudo-segmentsobtainedfrom

C

bythisprocedure.

Partition

C

 todisjointsubsets

C

1

,

C

2

,

. . . ,

C

ssothattwocurvesareinthesamesubsetifandonlyiftheirendpointsare

thesame.Notethats



t+21



.Fixtheorientationofeachcurvein

C

from Pi to Pjifi

<

j,andarbitrarilyifi

=

j.

Let fk denotethenumberoftouchingsinside

C

k and fk,l denotethenumberoftouchingsbetween

C

k and

C

l.Thenthe

totalnumberoftouchingsin

C

is

f

(

n

)

=



k fk

+



k<l fk,l

=



k O

(

|

C

k

|) +



k<l O

(

|

C

k

| + |

C

l

|) =

O

(

n

)

+



k

(

s

1

)

O

(

|

C

k

|) =

O

(

sn

)

=

O



t2n



,

wherethesecondequationfollowsfromLemma 5.

2

Noticethat incaseofagroundedintersecting familyofpseudo-segments,wehave s

=

t

+

1, so O

(

sn

)

=

O

(

tn

)

,which provesTheorem 4.

3. Quasi-gridsandtheiroccurrence 3.1. Notationsanddefinitions

We introduceseveralnotationsused inthe paper.Let g and h bea pairofdirected curves.If g touches theleft side of h,andthey havethesamedirectionatthetouching point,thenwrite g



h.(More precisely, let

γ

be acirclearound theintersection P witha smallenoughradius sothat itintersects both a andb twice, andthe diskdeterminedby

γ

is disjointfromalltheothercurves,andcontainsnootherintersectionsofa andb.Welabelthepointswhereg andh enters

γ

by g andh,andassignthelabelsgandhtothepointswheretheyexit.Wesaythattherightsideofg touchestheleft sideofh in P ifthecounter-clockwisecyclicorderoflabelson

γ

is ghhg.)Noticethatthisrelationisnotsymmetric,i.e.,

g



h



h



g.Ifg andh havedifferentdirectionsatthetouchingpoint(sothecounter-clockwisecyclicorderoflabelson

γ

isgghhorghhg),thenwrite g



h org



h dependingonwhichsideofh istouchedby g.Wesaythatc1 andc2 are g-touchequivalent iftheytouch g onthesamesideandinthesamedirection,i.e.,

(

g



c1

g



c2

)

or

(

g



c1

g



c2

)

or

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32 P. Györgyi et al. / Computational Geometry 67 (2018) 29–37

Fig. 1. A quasi-grid for the case gci. Swapping X,Y or A,B gives the other 3 cases.

For a directed curve g with points A and B that lie on the curve in this order, let A

−→

g B be the closed directed subcurve from A to B,and B

←−

g A will denotethe reversedirected subcurvefrom B to A.Thisnotation canbe iterated, e.g. ifP

h

g,then A

−→

g P

←−

h Q denotesthecurvewhichstartsfrom A

g,continueson g totheintersectionpoint P ,

then changestoh,andgoesonh inreverse directionuntilit endsin Q

h.When referringtoundirectedsubcurves,we use A g B.Sometimesthesenotationsarealsousedtodenotetheorderingofpointsonaparticularcurve.

As alreadydefined,

C(

A

,

B

)

isthesetofdirectedcurvesgoingfrom A to B.Foracurve c

C(

A

,

B

)

,letc

=

c

\ {

A

,

B

}

.

Forasetofcurves

C = {

c1

,

c2

,

. . .

ck

}

,let

C



= {

c1

,

c2

,

. . .

ck

}

.

The objectscalledquasi-gridsarethe maintoolofthispaper.Intuitively,thebelowdefinitionsaysthattheincidences ofaquasi-gridareexactlyasshowninFig. 1,withtheexceptionofthepoints X

,

Y

,

A and B —weallowthesetocoincide arbitrarily.

Definition6(Quasi-grid).Asetofcurves

C = {

c1

,

c2

,

. . . ,

ck

}

C(

A

,

B

)

formsaquasi-grid withrespecttoacurveg

C(

X

,

Y

)

if:

1.

C



∪ {

g

}

isanintersectingfamilyofpseudo-segments

2.

C

isg-touch-equivalentwithtouchingpoints P

i

=

g

ci

3. Pi,j

=

ci

cjisacrossingpoint

4. theorderingofpointsong is P1 g P2 g

. . .

g Pk

5. theorderingofpointsoncj( j

=

1

,

2

,

. . . ,

k)is

A

−→

cj P1,j cj

−→

P2,j cj

−→. . .

cj

−→

Pj−1,j cj

−→

Pj cj

−→

Pj,j+1 cj

−→. . .

cj

−→

Pj,k cj

−→

B

.

Anexampleforaquasi-gridcanbeseeninFig. 1.Throughoutthepaper(ifwedonotindicateitotherwise)weassume thattheindicesofthecurvesin

C

describetheorderoftheirtouchingpointsong.

3.2. Findingquasi-gridsincurveconfigurations

The goalofthissubsectionistoprove thatina familyofpseudo-segments, thesetof g-touchequivalent curveswith givenendpoints formaconstant numberofquasi-grids.Intuitively,Lemma 7 showsthat ina familyofpseudo-segments, the g-touchequivalentcurvesfrom

C(

A

,

B

)

(where A

=

B)canstillhavetwodistincttypes.Notethatthesetypescannotbe definedseparately,onlyinrelationtoeach other.InLemma 8,weestablishthatthecurvesineachtypeformaquasi-grid withrespectto g.Lemma 9examinesthecaseA

=

B.

Lemma7.Fixacurveg

C(

X

,

Y

)

andsupposethatc1 andc2 areg-touch-equivalentcurvesfrom

C(

A

,

B

)

withtouchingpoints P1andP2respectively,where A

=

B.NotethatP1andP2 dividec1andc2intotheirfirstandsecondparts.Supposefurtherthat

{

c

1

,

c2

,

g

}

isafamilyofpseudo-segments.Thenc1crossesc2atapointQ ,whichiseithertheintersectionofthefirstpartofc1with thesecondpartofc2,orviceversa:theintersectionofthesecondpartofc1withthefirstpartofc2.

Proof. Suppose(withoutlossofgenerality)that g



c1,g



c2,andthat P1precedes P2 ong.Considerthecloseddirected

curve



=

A

−→

c1 P1

g

−→

P2

c2

←−

A (curves withgrayhalointhemiddleandrightpartofFig. 2).Weshow that



isa Jordan-curve. Supposeforcontradictionthat A

−→

c1 P1 and A

c2

−→

P2 hasanintersectionpoint H

=

A (seetheleftpictureinFig. 2).

Since

{

c

1

,

c2

,

g

}

is afamilyofpseudo-segments,therecan benofurther intersectionsbetweenc1 andc2.Itfollowsthat





=

H

−→

c1 P1

g

−→

P2

c2

←−

H isaJordancurvethatseparatestheplane intoitsleftandright(shaded)sideregions.Noticethat

P1

c1

−→

B beginsintherightsideregion of



,while P2

c2

−→

B beginsintheleft sideregionby ourassumptions g



c1 and g



c2.Since P1

c1

−→

B

←−

c2 P2 isacontinuouscurvethat beginsandendsindifferentsidesof



,itmustcross



 inapoint

distinctfromH ;wearrivedatacontradiction. Thus

(

A

−→

c1 P1

)

∩(

A

c2

−→

P2

)

= {

A

}

,hence



isaJordan-curve.Bysimilarargumentasabove, P1

c1

−→

B

←−

c2 P2isacontinuous

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Fig. 2. Left: c1and c2cannot intersect before reaching g; middle and right: the possible configurations.

Fig. 3. Two quasi-grids with respect to g.

andP2 alreadyaccountfortheintersectionsbetweeng and

{

c1

c2

}

,theonlyremainingpossibilitiesarethatthecrossing

pointisasclaimed, i.e., Q

= (

A

−→

c2 P2

)

∩ (

P1

c1

−→

B

)

orQ

= (

A

−→

c1 P1

)

∩ (

P2

c2

−→

B

)

(seethemiddleandtherightpicture in Fig. 2).

2

Noticethattheabovelemmastatesthatthecurvec2meetsc1beforeitmeetsg ifandonlyifthefirstpartofc2 crosses

thesecondpartofc1andviceversa:c2 meetsg beforeitmeetsc1ifandonlyifthesecondpartofc2crossesthefirstpart

ofc1.Thisequivalencewillbeusedseveraltimesinthefollowinglemmas.

Lemma8.Letg

C(

X

,

Y

)

,andlet

H

beasetofg-touch-equivalentcurvesfrom

C(

A

,

B

)

,where A

=

B.If

H



∪ {

g

}

isafamilyof pseudo-segments,then

H

isthedisjointunionofatmosttwoquasi-gridswithrespecttog.

Proof. Wedealwiththecaseg



h forallh

H

,theotherthreecasesaresimilar.Leth

H

bethecurvethathasthefirst touchingpoint on X

−→

g Y amongthecurvesfrom

H

.Let

H

1

H

consistofh andthecurvesfrom

H

thatmeeth before

theymeetg.Let

H

2

:=

H \ H

1.Weprovethat

H

1and

H

2 arebothquasi-gridswithrespecttog.

Let

H

1

= {

h

=

h1

,

h2

,

. . . ,

h

}

andlet Pi

=

g

hi.Assumewithoutlossofgeneralitythat P1

g

−→

P2

g

−→. . .

−→

g P.Weshow

that

H

1isaquasi-gridwithrespectto g.(SeeFig. 3.)

Theproofisbyinductiononthenumberofcurvesin

H

1,for



=

1 thestatementistrivial.Lemma 7yieldsthestatement

for



=

2.

Weclaim that h crossesh1 between P1,−1 and B.By thedefinitionof

H

1 andLemma 7, P1, mustlie on P1

h1

−→

B.

Suppose forcontradictionthat the orderingon h1 is P1

h1

−→

P1,

h1

−→

P1,−1 (topleft ofFig. 4).Consider the closedJordan

curvec

=

P1 g

−→

P−1 h−1

←−

P1,−1 h1

←−

P1.(Therightsideregionofc isshaded.)

Notice that A and B are on theleft side ofc. To seethis, considerthat c ismade up of threecurve segments, and therecanbenofurtherintersectionsamongthesethreecurves,soh1 andc

\ (

P1

h1

−→

P1,−1

)

aredisjoint.Sincethetype of

touchingat P1 is g



h1,wecanseethat

(

A

h1

−→

P1

)

\

P1andconsequentlypoint A inparticularliesintheleftsideregion

ofc.Asimilarargumentforh−1showsthat B isalsointheleftsideregion.

Sinceh isalreadycrossingc once atP1,,ithastocrossitonemoretime,becauseitsendpoints A and B areonthe

samesideofc.Sinceh touchesg outsidec anditalreadyhasan intersectionwithh1,itwillcross P−1

h−1

←−

P1,−1.Now hhasenteredtherightsideof P1,−1

h−1

−→

B

←−

h1 P1,−1 (theregionwiththelinepattern),whilePisontheleftside(since

g



h−1).Sohwouldhavetocrossh1orh−1 onemoretime,whichisacontradiction.

If



=

3,thenconsiderthecurve c

=

P2

g

−→

P3 h3

−→

B

←−

h1 P1,3 h1

←−

P1,2 h2

−→

P2 (see thetop rightofFig. 4).Sinceh2 andh3

(7)

34 P. Györgyi et al. / Computational Geometry 67 (2018) 29–37

Fig. 4. Top left: if P1,is on P1 h1

−→P1,−1; top right: the case|H1| =  =3; bottom: the case≥4.

(

P3

h3

−→

B

)

).Thus,h2andh3crosseachotheratapoint P2,3

= (

P1,3

h3

−→

P3

)

∩ (

P2

h2

−→

B

)

.Byinduction,thepointsonh1and h2 arealsointherequiredorder.

For



4 theinductionis usedboth forh1

,

h2

,

. . . ,

h−1 andh1

,

h3

,

h4

,

. . . ,

h.We onlyneed to showthat h2 andh

crosseachotheratapoint P2, whichsatisfiesourorderingconditions.Indeed,ontherightsideofthecurve

c

=

P2 g

−→

P3 h3

−→

P3, h3

−→

B

←−

h1 P1, h1

←−

P1,2 h2

−→

P2

,

thereisacrossingP2,

= (

P1, h

−→

P3,

)

∩ (

P2 h2

−→

B

)

:thisshowsthattheorderingonhiscorrect(seethebottomofFig. 4).

Asimilarargumentshowsthattheorderingofpointsonh2 iscorrect,oneneedstoconsidertherightsideofthefollowing

closedcurve: P1,−1 h−1

−→

P2,−1 h−1

−→

P−1, h−1

−→

B

←−

h1 P1, h1

←−

P1,−1

.

We showthat

H

2 behaves similarly.Let

H

2

= {

h1

,

h2

,

. . . ,

hm

}

andlet Pi

=

g

hi.Again, supposethat P1

g

−→

P2

−→. . .

g

g

−→

Pm (seeFig. 3).ByLemma 7,h1 mustcrossh

=

h1atapoint Q

A

h1

−→

P1,sinceh1

/

H

1.NowconsidertheJordancurve e

=

P1

g

−→

P1 h  1

−→

Q

−→

h1 P1. Again,it iseasy tocheck that e separates A from Pj for j

2.Consider acurve hj

(

j

2

)

. It

cannotmeeth1 beforemeeting g sincehj

/

H

1.Thus A

hj

−→

Pj mustcrosse somewhereon P1 h  1

−→

Q .Wehavereducedthis problemtotheprevioussituationwithh1actingash1,so

H

2alsoformsaquasi-gridwithrespecttog.

2

Intheproof ofthenext lemma,weusethetouchinggraph ofacurve familyofpseudo-segments.Let

H

beafamilyof pseudo-segments.Thetouchinggraph of

H

isGH

= (

V

,

E

)

withV

=

H

andE

= {(

a

,

b

)

:

a



b

}

.Thestatementofthislemma isalmostidenticaltothepreviousone,butconsidersthecasewhentheendpointsofthequasi-gridcoincide.Inthiscase, wecannotprovethatthecurvesaretheunionofatmosttwoquasi-grids,butwecanstillboundthenumberofquasi-grids byaconstant.

Lemma9.Let g

C(

X

,

Y

)

,and

H

is asetof g-touch-equivalentcurvesfrom

C(

A

,

A

)

.If

H



∪ {

g

}

isanintersectingfamilyof

pseudo-segments,then

H

isthedisjointunionofatmost12quasi-gridswithrespecttog.

Proof. Again, weonly dealwiththe case g



h for all h

H

.The firstclaim isthat anyh

H

touchesatmost2other curvesin

H

.Sinceh isaJordancurve,itseparatestheplane intotworegions,one ofthesecontains g;denotethisregion by Rg, andtheother by Rn (see Fig. 5, Rn hasa linepattern). Observethat no curve in

H

touching h can enter Rn as

such acurvecannot touch g.Let P

=

g

h.We provethatthereisatmostonecurvein

H

thattouches A

−→

h P .Suppose for contradiction that curves h1

,

h2

H

are both touching A

h

−→

P at points T1 and T2 respectively, withthe ordering A

−→

h T1

h

−→

T2

h

−→

P .Let Qi

=

hi

g.Notethat A liesontheleftsideofthecurvec

=

Q1

g

−→

P

←−

h T1 h1 Q1 sinceg



h.By

anearlierobservation,h2isdisjointfromtheopenregionRn,whichistherightsideofh —soh2touchestheleftsideofh

in T2,i.e.,h2



h orh2



h.Itfollowsthatboth A

h2

−→

T2 andT2

h2

−→

A mustcrossc,butthiscrossingcanonlyhappenalong

(8)

Fig. 5. Curve h touches at most two other curves inH.

whichcontradictsthebasicpropertiesofanintersecting familyofpseudo-segments.Asimilarargumentshowsthatthere isatmostonecurvein

H

thattouchesP

−→

h A.

Consider the touching graph GH. Ourfirst observationimplies that the maximal degree in GH is2, thus by Brooks’ theorem[15],GH is3-colorable.Itissufficienttoprovethateachcolorclassisthedisjointunionofatmost4quasi-grids withrespecttog.

Let

H

0

H

beacolorclass;consequently,itcannotcontainatouchingpairofcurves,i.e.,thecurvesin

H

arepairwise

intersecting. Letk

= |

H

0

|

.Inthisparagraph, an endingofa directed curve refers toone ofthe endings ofits undirected

version. Consider thecyclicorderofthe curve endingsof

H

0 around A: x1

,

x2

,

. . . ,

x2k.Eachcurve appears exactlytwice

in thissequence. Eachpair ofcurves in

H

0 crosses, hencexi and xk+i belong to the samecurve foreach i

∈ {

1

,

. . . ,

k

}

.

Thereforewe maydilate A to twopoints A1 and A2 suchthat endings x1

,

. . . ,

xk are at A1 andendings xk+1

,

. . . ,

x2k are

at A2.Now

H

0 canbeconsidered asafamilyof A1 A2 curves,whichistheunionof

H

12containing A1

−→

A2 curves

and

H

21 containing A2

−→

A1 curves.AccordingtoLemma 8,both

H

12and

H

21aretheunionofatmosttwoquasi-grids

withrespecttog.

2

TheaboveLemmasalsoimplythefollowingone:

Lemma10.LetA

,

B

,

C

,

D benotnecessarilydistinctpointsontheplane.Letg

C(

A

,

B

)

,andlet

C

0

C(

C

,

D

)

beafinitecurvefamily suchthat

{

g

}

C

0isanintersectingfamilyofpseudo-segmentswithallh

C

0touchingg.Then

C

0isthedisjointunionofatmost48 quasi-gridswithrespecttog.

Proof.

C

0 isthedisjointunionofatmostfourg-touchingequivalenceclasses(g



h,g



h,h



g andh



g).ByLemmas 8

and9,eachsuchclasscanbedecomposedintoatmost12quasi-grids.

2

4. Touchingquasi-gridswithexternalcurves

Lemma11.Letg beanycurvein

C(

A

,

B

)

andlet

H = {

h1

,

h2

,

h3

}

C(

C

,

D

)

beaquasi-gridwithrespectto g (possiblyapart ofalargerquasi-grid).Supposethatforacurveg

C(

A

,

B

)

,thesetoffivecurves

{

g

,

g

,

h1

,

h2

,

h3

}

isanintersectingfamilyof pseudo-segmentsandgtouchesthemiddlecurveh2

H

.Thengmustalsotouchatleastonemoreamong

{

g

,

h1

,

h3

}

.

Proof. Supposethatg



hi

(

i

=

1

,

2

,

3

)

,theothercasesaresimilar.Thedefinitionofquasi-gridsenumeratesallintersections

betweenthefourcurvesh1

,

h2

,

h3 andg.ItfollowsthatthebordersofthefacesintherightsideofC

h1

−→

P1 g

−→

P3 h3

−→

D

←−

h1 P1,3 h3

←−

C aredetermined(seethefacesmarkedwithencirclednumbersinFig. 6).Noticethatsome(orall)of A

,

B

,

C and D maycoincide,sotheotherfacesareunknown.Let 1 betherightsideofC

−→

h1 P1,2

h2

←−

C .Inthesamemanner,weassign numbers 2

5 tosomeotherfacesaswell,seeFig. 6.

Supposeforcontradictionthat g crossesh1,h3 andg.Weneedthefollowingclaimtoproceedwithourproof. Claim12.Thecurvegcannotenterregion 5 .

Proof. If gpassesthrough 5 ,then—sinceittouchesh2 — itmustcrossboth P1,2

h1

−→

P1,3 and P1,3

h3

−→

P2,3.Sincethere

can beno moreintersectionswithh1 orh3,thecurve g cannot passthrough theclosedcurve c

=

C

h2

−→

D

←−

h1 P1,3

h3

←−

C ,

thereforeitcannotmeetg(seeFig. 6).

2

Since g mustcrossh1,ithastoentereitherregion 1 or 4 (byClaim 12itcannotcross P1,2

h1

−→

P1,3).Ifitenters 1 ,

then—sinceithascrossedh1,oneofitsendpoints A orB hastobeontheborderof 1 ,thuseither A

=

C or B

=

C .Ifg

enters 4 ,thenbyClaim 12,oneoftheendpointsisD,so A

=

D orB

=

D.Thecurvegalsoneedstocrossh3,soitenters

(9)

36 P. Györgyi et al. / Computational Geometry 67 (2018) 29–37

Fig. 6. The figures for various possible equalities among A,B,C and D.

Fig. 7. A 5-element quasi-gridH.

If g enters 1 and 3 , then A

=

C

=

B.Since 1 and 3 are on thesameside ofthe closedcurve g

C(

A

,

A

)

,the curve

(

g

)



C



(

A

,

A

)

crosses g atanevennumberofpoints,we arrivedata contradiction.Thecasewhen g enters 2

and 4 isidenticalifweswaptheroleofC and D.

If g enters 1 and 2 , thenthe endpointsof g

C(

A

,

B

)

are C and D.If A

=

D and B

=

C , thenthe closedcurves

B

−→

h1 P1

g

−→

B and A

−→

g P1

h1

−→

A must cross each other at an even number of points, so there is an intersection point distinct from P1.Notethat h1 andg aremembers ofanintersecting familyofpseudo-segments (since

H

isa quasi-grid

withrespectto g),sotheintersectionmustbeattheirendpoints,thus A

=

B.Consequently,ifg enters 1 and 2 , then either A

=

B

=

C

=

D or A

=

C and B

=

D aretwo distinctpoints (seethe bottomofFig. 6).Let R1 betheregion tothe

left of A

−→

h2 P2

g

←−

A (sparselydotted)and R2 betheleftsideofB

g

←−

P2

h2

−→

B (denselydotted).Noticethat g startsin R1

andendsinR2,tworegionsthatareguaranteedtobedisjointapartfromP2

,

A and B.Sinceitcannotcrossh2,itcrosses

both g

1andg2,where g1

=

A

g

−→

P2and g2

=

P2

g

−→

B.Thisisacontradictionsince ghastocross g exactlyonce. If genters 3 and 4 , then A

=

C andB

=

D bythesameargumentasintheprevious case.Let Aand Bbepoints on g closetoitsstartingandendpoint A and B,sothat therearenotouchingsorcrossingsong betweenA and Aand between B andB.Notethat gcannotcrossh2 becausetheyneedtotouch.ThustheboundaryofR1 canonlybecrossed

on g1,andboth AandBlie outsideR1 (theyarein 3 and 4 respectively),sothenumberofintersectionsbetweeng1

and

(

g

)

iseven.ThesameargumentholdsforR

2 andg2,sothenumberofintersectionsbetweengand

(

g

)

iseven—

acontradiction.

2

Thenextlemmademonstratesourintuitiveclaimthattouchingthemembersofalargequasi-gridbytwocurvesisnot possibleinsideanintersectingfamilyofpseudo-segments.

Lemma13.Let

H

beasetofatleast5curvesfrom

C(

A

,

B

)

,whereA andB maycoincide.Letg1

,

g2betwocurvessuchthat

H ∪

{

g1

,

g2

}

formafamilyofpseudo-segments.Then

H

cannotformaquasi-gridwithrespecttobothg1andg2.

Proof. Supposeforcontradictionthat

H

isaquasi-gridwithrespectto g1 andg2 simultaneously.Let

H



= {

h1

,

h2

,

. . . ,

h5

}

(10)

Thecurve g2 cannot haveanypointsinaregion whichisenclosed byonlycurvesfrom

H

:itcannot leavetheregion

sinceitcannotcrossanyof

H

,andeveryregionisboundedbyatmostfourofthe

H

curves,soatleastonecurvewould remainuntouchableforg2.

Consequently, g2 hasto touchh2,h3 andh4 intheregions enclosed by g1

,

hi andhi+1

(

i

=

1

,

2

,

3

,

4

)

(see theshaded

regionsinFig. 7).Since g2 canmeetg1atmostonce,itcanvisitonlyoneoftheseregions,soatleastoneofh2

,

h3andh4

willremainuntouchable—wearrivedatacontradiction.

2

Acknowledgements

We thank János Pach andGéza Tóth for suggestingthe original problem, for the encouragementand for the fruitful discussions.Wethankananonymous refereeforseveralremarksthatimprovedthepresentationofthepaper.Thisresearch wassupportedby theHungarianScientificResearchFund–OTKA,K109240.ThethirdauthorwasalsosupportedbyNWO grantno.024.002.003.

References

[1]M.Ajtai,V.Chvátal,M.M.Newborn,E.Szemerédi,Crossing-FreeSubgraphs,North-HollandMathematicsStudies,vol. 60,1982,pp. 9–12.

[2]F.T.Leighton,ComplexityIssuesinVLSI:OptimalLayoutsfortheShuffle-ExchangeGraphandOtherNetworks,MITPress,Cambridge,MA,USA,1983. [3] E.Szemerédi,W.T.TrotterJr,Extremalproblemsindiscretegeometry,Combinatorica3 (3–4)(1983)381–392,http://dx.doi.org/10.1007/BF02579194. [4] K.Kedem,R.Livne,J.Pach,M.Sharir,Onthe unionofJordanregionsandcollision-freetranslationalmotionamidstpolygonalobstacles,Discrete

Comput.Geom.1 (1)(1986)59–71,http://dx.doi.org/10.1007/BF02187683.

[5] H.Tamaki, T.Tokuyama, Howtocutpseudoparabolas into segments,DiscreteComput. Geom.19 (2) (1998)265–290, http://dx.doi.org/10.1007/ PL00009345.

[6] B.Aronov, M.Sharir,Cuttingcirclesinto pseudo-segmentsand improvedboundsfor incidences,DiscreteComput.Geom.28 (4) (2002)475–490, http://dx.doi.org/10.1007/s00454-001-0084-1.

[7] P.K.Agarwal,E.Nevo,J.Pach,R.Pinchasi,M.Sharir,S.Smorodinsky,Lensesinarrangementsofpseudo-circlesandtheirapplications,J.ACM51 (2) (2004)139–186,http://dx.doi.org/10.1145/972639.972641.

[8] A.Marcus,G.Tardos,Intersectionreversesequencesandgeometricapplications,J.Comb.Theory,Ser.A113 (4)(2006)675–691,http://dx.doi.org/ 10.1016/j.jcta.2005.07.002.

[9] J.Fox,J.Pach,C.D.Tóth,Intersectionpatternsofcurves,J.Lond.Math.Soc.83(2011)389–406,http://dx.doi.org/10.1112/jlms/jdq087.

[10] R.B. Richter,C.Thomassen,IntersectionsofcurvesystemsandthecrossingnumberofCC5,DiscreteComput.Geom.13 (1)(1995)149–159, http://dx.doi.org/10.1007/BF02574034.

[11]J. Pach,N.Rubin,G.Tardos,Beyondthe Richter–Thomassenconjecture,in:ProceedingsoftheTwenty-SeventhAnnualACM-SIAMSymposiumon DiscreteAlgorithms,2016,pp. 957–968.

[12]J.Fox,F.Frati,J.Pach,R.Pinchasi,Crossingsbetweencurveswithmanytangencies,in:AnIrregularMind,in:BolyaiSoc.Math.Stud.,vol. 21,2010, pp. 251–260.

[13] J.Pach,Privatecommunication,2015.

[14] J.Pach,M.Sharir,OnverticalvisibilityinarrangementsofsegmentsandthequeuesizeintheBentley–Ottmann linesweepingalgorithm,SIAMJ. Comput.20 (3)(1991)460–470,http://dx.doi.org/10.1137/0220029.

[15] R.L. Brooks,Oncolouringthe nodesofanetwork, Math. Proc.Camb. Philos.Soc.Math. Phys. Sci.37(1941)194–197, http://dx.doi.org/10.1017/ S030500410002168X.

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