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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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,HungarybMTA-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,whereϕ
isinjectiveon[0,
1)
andϕ
(
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,andthediskdeterminedbyγ
isdisjointfromalltheothercurves,andcontainsnootherintersections ofa andb.Labeltheintersectionpoints ofγ
andthetwocurveswiththenameofthecurve.Wesaythat a andb crossin P ifthecyclicalpermutationoflabelsaround
γ
isabab,anda andb touch in P ifthecyclical permutationoflabelsisaabb.Inafamilyofcurves,let X bethesetofcrossingsandT bethesetoftouchings.
TheRichter–Thomassen conjecture[10]statesthatgivenacollectionofn pairwiseintersectingclosedJordancurvesin generalpositionintheplane,thenumberofcrossingsisatleast
(
1−
o(
1))
n2.AproofoftheRichter–ThomassenconjecturehasrecentlybeenpublishedbyPachetal.[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
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.SupposethatanypairofcurvesinC
intersectexactlyonce.Thenthenumberof touchingsinC
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 eachcurveinC
hasoneendpointon g andtherestofthecurveisintheexteriorofg.Thecollectionisdoublegrounded iftherearedisjointclosedJordancurvesg1 andg2suchthateachcurvec
∈
C
hasoneendpointong1 andtheotherendpointon 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-groundedx-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 curveinC
.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.
Beforestatingthekey lemma,we introducesomenotations.Thenotation g
h meansthatcurves g andh toucheach other.If A and B are(notnecessarilydistinct)points ontheplane,thenC(
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,andC
1andC
2befinitedisjointcurvefamiliesfromC(
A,
B)
andC(
C,
D)
,respectively.IfC
1∪
C
2isanintersectingfamilyofpseudo-segments,then1. 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|)
instancesofc1c2 touchings.Let K bealarge constant.Without lossofgenerality, wecan supposethat eachcurve of
C
i touchesatleast K curvesof
C
j.Toseethis,considerfirstthebipartite graphG withvertexsetC
1∪
C
2,wheretheedges correspondtothec1c2touchings(c1
∈
C
1 andc2∈
C
2).IfG hasverticesofdegreelessthan K ,thendeletethoseverticesandtheincidentedges.IteratethisprocedureuntiltheminimumdegreeisatleastK orthegraphisempty.IfG hadatleastK
|
C
1∪
C
2|
edges,thenthisprocedurecannotresultinanemptygraph.
Let g
∈
C
1 be an arbitrarycurve. ByLemma 10, there is a quasi-grid withrespect to g formed by atleast K/
48>
3curves.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-gridQ
withrespecttobothh andh.Therefore,bychoosingK
≥
4·
482+
1,thequasi-gridQ
canbeforcedtocontainatleastfivecurves.Thisisa contradic-tionbyLemma 13.2
NextweshowhowLemma 5impliesTheorem 2.Lett
=
tC.Proof ofTheorem 2. Consider theplanargraphdrawingthat correspondsto
C
.Letthe facesofthisplanargraphdrawing thatcontainanendpointofatleastonecurveinC
be:F1,
F2,
. . . ,
Ft.Fori
=
1,
2,
. . . ,
t,let Pi beanarbitrarypointintheinteriorof Fi notincidentto anycurveinC
.Eachcurve endpointinside Ficanbeconnectedto Piwithoutaddinganyintersectionsbetweenthecurvesof
C
withtheexceptionofPi.LetC
bethefamilyofpseudo-segmentsobtainedfrom
C
bythisprocedure.Partition
C
todisjointsubsetsC
1,
C
2,
. . . ,
C
ssothattwocurvesareinthesamesubsetifandonlyiftheirendpointsarethesame.Notethats
≤
t+21.FixtheorientationofeachcurveinC
from Pi to Pjifi<
j,andarbitrarilyifi=
j.Let fk denotethenumberoftouchingsinside
C
k and fk,l denotethenumberoftouchingsbetweenC
k andC
l.Thenthetotalnumberoftouchingsin
C
isf
(
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, letγ
be acirclearound theintersection P witha smallenoughradius sothat itintersects both a andb twice, andthe diskdeterminedbyγ
is disjointfromalltheothercurves,andcontainsnootherintersectionsofa andb.Welabelthepointswhereg andh entersγ
by g andh,andassignthelabelsgandhtothepointswheretheyexit.Wesaythattherightsideofg touchestheleft sideofh in P ifthecounter-clockwisecyclicorderoflabelsonγ
is ghhg.)Noticethatthisrelationisnotsymmetric,i.e.,g
hhg.Ifg andh havedifferentdirectionsatthetouchingpoint(sothecounter-clockwisecyclicorderoflabelsonγ
isgghhorghhg),thenwrite gh org
h dependingonwhichsideofh istouchedby g.Wesaythatc1 andc2 are g-touchequivalent iftheytouch g onthesamesideandinthesamedirection,i.e.,
(
gc1∧
gc2)
or(
g c1∧
g c2)
or32 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}
,letC
= {
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-segments2.
C
isg-touch-equivalentwithtouchingpoints Pi
=
g∩
ci3. Pi,j
=
ci∩
cjisacrossingpoint4. theorderingofpointsong is P1 g P2 g
. . .
g Pk5. theorderingofpointsoncj( j
=
1,
2,
. . . ,
k)isA
−→
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-equivalentcurvesfromC(
A,
B)
withtouchingpoints P1andP2respectively,where A=
B.NotethatP1andP2 dividec1andc2intotheirfirstandsecondparts.Supposefurtherthat{
c1
,
c2,
g}
isafamilyofpseudo-segments.Thenc1crossesc2atapointQ ,whichiseithertheintersectionofthefirstpartofc1with thesecondpartofc2,orviceversa:theintersectionofthesecondpartofc1withthefirstpartofc2.Proof. Suppose(withoutlossofgenerality)that g
c1,gc2,andthat P1precedes P2 ong.Considerthecloseddirectedcurve
=
A−→
c1 P1g
−→
P2c2
←−
A (curves withgrayhalointhemiddleandrightpartofFig. 2).Weshow thatisa Jordan-curve. Supposeforcontradictionthat A
−→
c1 P1 and Ac2
−→
P2 hasanintersectionpoint H=
A (seetheleftpictureinFig. 2).Since
{
c1
,
c2,
g}
is afamilyofpseudo-segments,therecan benofurther intersectionsbetweenc1 andc2.Itfollowsthat=
H−→
c1 P1g
−→
P2c2
←−
H isaJordancurvethatseparatestheplane intoitsleftandright(shaded)sideregions.NoticethatP1
c1
−→
B beginsintherightsideregion of,while P2
c2
−→
B beginsintheleft sideregionby ourassumptions gc1 and gc2.Since P1c1
−→
B←−
c2 P2 isacontinuouscurvethat beginsandendsindifferentsidesof,itmustcross
inapoint
distinctfromH ;wearrivedatacontradiction. Thus
(
A−→
c1 P1)
∩(
Ac2
−→
P2)
= {
A}
,henceisaJordan-curve.Bysimilarargumentasabove, P1
c1
−→
B←−
c2 P2isacontinuousFig. 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}
,theonlyremainingpossibilitiesarethatthecrossingpointisasclaimed, i.e., Q
= (
A−→
c2 P2)
∩ (
P1c1
−→
B)
orQ= (
A−→
c1 P1)
∩ (
P2c2
−→
B)
(seethemiddleandtherightpicture in Fig. 2).2
Noticethattheabovelemmastatesthatthecurvec2meetsc1beforeitmeetsg ifandonlyifthefirstpartofc2 crosses
thesecondpartofc1andviceversa:c2 meetsg beforeitmeetsc1ifandonlyifthesecondpartofc2crossesthefirstpart
ofc1.Thisequivalencewillbeusedseveraltimesinthefollowinglemmas.
Lemma8.Letg
∈
C(
X,
Y)
,andletH
beasetofg-touch-equivalentcurvesfromC(
A,
B)
,where A=
B.IfH
∪ {
g}
isafamilyof pseudo-segments,thenH
isthedisjointunionofatmosttwoquasi-gridswithrespecttog.Proof. Wedealwiththecaseg
h forallh∈
H
,theotherthreecasesaresimilar.Leth∈
H
bethecurvethathasthefirst touchingpoint on X−→
g Y amongthecurvesfromH
.LetH
1⊆
H
consistofh andthecurvesfromH
thatmeeth beforetheymeetg.Let
H
2:=
H \ H
1.WeprovethatH
1andH
2 arebothquasi-gridswithrespecttog.Let
H
1= {
h=
h1,
h2,
. . . ,
h}
andlet Pi=
g∩
hi.Assumewithoutlossofgeneralitythat P1g
−→
P2g
−→. . .
−→
g P.Weshowthat
H
1isaquasi-gridwithrespectto g.(SeeFig. 3.)Theproofisbyinductiononthenumberofcurvesin
H
1,for=
1 thestatementistrivial.Lemma 7yieldsthestatementfor
=
2.Weclaim that h crossesh1 between P1,−1 and B.By thedefinitionof
H
1 andLemma 7, P1, mustlie on P1h1
−→
B.Suppose forcontradictionthat the orderingon h1 is P1
h1
−→
P1,h1
−→
P1,−1 (topleft ofFig. 4).Consider the closedJordancurvec
=
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
\ (
P1h1
−→
P1,−1)
aredisjoint.Sincethetype oftouchingat P1 is g
h1,wecanseethat(
Ah1
−→
P1)
\
P1andconsequentlypoint A inparticularliesintheleftsideregionofc.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,−1h−1
−→
B←−
h1 P1,−1 (theregionwiththelinepattern),whilePisontheleftside(sinceg
h−1).Sohwouldhavetocrossh1orh−1 onemoretime,whichisacontradiction.If
=
3,thenconsiderthecurve c=
P2g
−→
P3 h3−→
B←−
h1 P1,3 h1←−
P1,2 h2−→
P2 (see thetop rightofFig. 4).Sinceh2 andh334 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.
(
P3h3
−→
B)
).Thus,h2andh3crosseachotheratapoint P2,3= (
P1,3h3
−→
P3)
∩ (
P2h2
−→
B)
.Byinduction,thepointsonh1and h2 arealsointherequiredorder.For
≥
4 theinductionis usedboth forh1,
h2,
. . . ,
h−1 andh1,
h3,
h4,
. . . ,
h.We onlyneed to showthat h2 andhcrosseachotheratapoint 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.LetH
2= {
h1,
h2,
. . . ,
hm}
andlet Pi=
g∩
hi.Again, supposethat P1g
−→
P2−→. . .
gg
−→
Pm (seeFig. 3).ByLemma 7,h1 mustcrossh=
h1atapoint Q∈
Ah1
−→
P1,sinceh1∈
/
H
1.NowconsidertheJordancurve e=
P1g
−→
P1 h 1−→
Q−→
h1 P1. Again,it iseasy tocheck that e separates A from Pj for j≥
2.Consider acurve hj(
j≥
2)
. Itcannotmeeth1 beforemeeting g sincehj
∈
/
H
1.Thus Ahj
−→
Pj mustcrosse somewhereon P1 h 1−→
Q .Wehavereducedthis problemtotheprevioussituationwithh1actingash1,soH
2alsoformsaquasi-gridwithrespecttog.2
Intheproof ofthenext lemma,weusethetouchinggraph ofacurve familyofpseudo-segments.Let
H
beafamilyof pseudo-segments.Thetouchinggraph ofH
isGH= (
V,
E)
withV=
H
andE= {(
a,
b)
:
ab}
.Thestatementofthislemma isalmostidenticaltothepreviousone,butconsidersthecasewhentheendpointsofthequasi-gridcoincide.Inthiscase, wecannotprovethatthecurvesaretheunionofatmosttwoquasi-grids,butwecanstillboundthenumberofquasi-grids byaconstant.Lemma9.Let g
∈
C(
X,
Y)
,andH
is asetof g-touch-equivalentcurvesfromC(
A,
A)
.IfH
∪ {
g}
isanintersectingfamilyofpseudo-segments,then
H
isthedisjointunionofatmost12quasi-gridswithrespecttog.Proof. Again, weonly dealwiththe case g
h for all h∈
H
.The firstclaim isthat anyh∈
H
touchesatmost2other curvesinH
.Sinceh isaJordancurve,itseparatestheplane intotworegions,one ofthesecontains g;denotethisregion by Rg, andtheother by Rn (see Fig. 5, Rn hasa linepattern). Observethat no curve inH
touching h can enter Rn assuch acurvecannot touch g.Let P
=
g∩
h.We provethatthereisatmostonecurveinH
thattouches A−→
h P .Suppose for contradiction that curves h1,
h2∈
H
are both touching Ah
−→
P at points T1 and T2 respectively, withthe ordering A−→
h T1h
−→
T2h
−→
P .Let Qi=
hi∩
g.Notethat A liesontheleftsideofthecurvec=
Q1g
−→
P←−
h T1 h1 Q1 sincegh.Byanearlierobservation,h2isdisjointfromtheopenregionRn,whichistherightsideofh —soh2touchestheleftsideofh
in T2,i.e.,h2
h orh2 h.Itfollowsthatboth Ah2
−→
T2 andT2h2
−→
A mustcrossc,butthiscrossingcanonlyhappenalongFig. 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.,thecurvesinH
arepairwiseintersecting. Letk
= |
H
0|
.Inthisparagraph, an endingofa directed curve refers toone ofthe endings ofits undirectedversion. Consider thecyclicorderofthe curve endingsof
H
0 around A: x1,
x2,
. . . ,
x2k.Eachcurve appears exactlytwicein 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 areat A2.Now
H
0 canbeconsidered asafamilyof A1 A2 curves,whichistheunionofH
12containing A1−→
A2 curvesand
H
21 containing A2−→
A1 curves.AccordingtoLemma 8,bothH
12andH
21aretheunionofatmosttwoquasi-gridswithrespecttog.
2
TheaboveLemmasalsoimplythefollowingone:
Lemma10.LetA
,
B,
C,
D benotnecessarilydistinctpointsontheplane.Letg∈
C(
A,
B)
,andletC
0⊂
C(
C,
D)
beafinitecurvefamily suchthat{
g}
∪
C
0isanintersectingfamilyofpseudo-segmentswithallh∈
C
0touchingg.ThenC
0isthedisjointunionofatmost48 quasi-gridswithrespecttog.Proof.
C
0 isthedisjointunionofatmostfourg-touchingequivalenceclasses(gh,g h,hg andh g).ByLemmas 8and9,eachsuchclasscanbedecomposedintoatmost12quasi-grids.
2
4. Touchingquasi-gridswithexternalcurves
Lemma11.Letg beanycurvein
C(
A,
B)
andletH = {
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-gridsenumeratesallintersectionsbetweenthefourcurvesh1
,
h2,
h3 andg.ItfollowsthatthebordersofthefacesintherightsideofCh1
−→
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 P1,2h2
←−
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 P1,3h3
−→
P2,3.Sincetherecan beno moreintersectionswithh1 orh3,thecurve g cannot passthrough theclosedcurve c
=
Ch2
−→
D←−
h1 P1,3h3
←−
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 .Ifgenters 4 ,thenbyClaim 12,oneoftheendpointsisD,so A
=
D orB=
D.Thecurvegalsoneedstocrossh3,soitenters36 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 2and 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 closedcurvesB
−→
h1 P1g
−→
B and A−→
g P1h1
−→
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 (sinceH
isa quasi-gridwithrespectto 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 totheleft of A
−→
h2 P2g
←−
A (sparselydotted)and R2 betheleftsideofBg
←−
P2h2
−→
B (denselydotted).Noticethat g startsin R1andendsinR2,tworegionsthatareguaranteedtobedisjointapartfromP2
,
A and B.Sinceitcannotcrossh2,itcrossesboth g
1andg2,where g1
=
Ag
−→
P2and g2=
P2g
−→
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 canonlybecrossedon g1,andboth AandBlie outsideR1 (theyarein 3 and 4 respectively),sothenumberofintersectionsbetweeng1
and
(
g)
iseven.ThesameargumentholdsforR2 andg2,sothenumberofintersectionsbetweengand
(
g)
iseven—acontradiction.
2
Thenextlemmademonstratesourintuitiveclaimthattouchingthemembersofalargequasi-gridbytwocurvesisnot possibleinsideanintersectingfamilyofpseudo-segments.
Lemma13.Let
H
beasetofatleast5curvesfromC(
A,
B)
,whereA andB maycoincide.Letg1,
g2betwocurvessuchthatH ∪
{
g1,
g2}
formafamilyofpseudo-segments.ThenH
cannotformaquasi-gridwithrespecttobothg1andg2.Proof. Supposeforcontradictionthat
H
isaquasi-gridwithrespectto g1 andg2 simultaneously.LetH
= {
h1,
h2,
. . . ,
h5}
Thecurve g2 cannot haveanypointsinaregion whichisenclosed byonlycurvesfrom
H
:itcannot leavetheregionsinceitcannotcrossanyof
H
,andeveryregionisboundedbyatmostfouroftheH
curves,soatleastonecurvewould remainuntouchableforg2.Consequently, g2 hasto touchh2,h3 andh4 intheregions enclosed by g1
,
hi andhi+1(
i=
1,
2,
3,
4)
(see theshadedregionsinFig. 7).Since g2 canmeetg1atmostonce,itcanvisitonlyoneoftheseregions,soatleastoneofh2
,
h3andh4willremainuntouchable—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.
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