On a Tree and a Path with no Geometric Simultaneous Embedding

42 Pages·0.56 MB·English

by  Patrizio Angelini

On a Tree and a Path with no Geometric Simultaneous
Embedding
P.Angelini†,M.Geyer‡,M.Kaufmann‡andD.Neuwirth‡
†DipartimentodiInformaticaeAutomazione–UniversitàRomaTre,Italy
[email protected]
‡Wilhelm-Schickard-InstitutfürInformatik–UniversitätTübingen,Germany
geyer/mk/[email protected]
0
1
0 Abstract. Two graphs G1 = (V,E1) and G2 = (V,E2) admit a geomet-
2 ric simultaneous embedding if there exists a set of points P and a bijection
M : P → V thatinduceplanarstraight-lineembeddingsbothforG1 andfor
n
G2.Whileitisknownthattwocaterpillarsalwaysadmitageometricsimultane-
a
ousembeddingandthattwotreesnotalwaysadmitone,thequestionaboutatree
J
andapathisstillopenandisoftenregardedasthemostprominentopenproblem
4
inthisarea.Weanswerthisquestioninthenegativebyprovidingacounterex-

ample.Additionally,sincethecounterexampleusesdisjointedgesetsforthetwo
]
G graphs, we also negatively answer another open question, that is, whether it is
C possibletosimultaneouslyembedtwoedge-disjointtrees.Asafinalresult,we
studythesameproblemwhensomeconstraintsonthetreeareimposed.Namely,
.
s weshowthatatreeofdepth2andapathalwaysadmitageometricsimultaneous
c
embedding.Infact,suchastrongconstraintisnotsofarfromclosingthegapwith
[
theinstancesnotadmittinganysolution,asthetreeusedinourcounterexample

1 hasdepth4.
v
5
5
1 Introduction
5
0
1. Embeddingplanargraphsisawell-establishedfieldingraphtheoryandalgorithmswith
0 agreatvarietyofapplications.KeystonesinthisfieldaretheworksofThomassen[15],
0 ofTutte[16],andofPachandWenger[14],dealingwithplanarandconvexrepresenta-
1
tionsofgraphsintheplane.
:
v Since recently, motivated by the need of contemporarily represent several differ-
Xi ent relationships among the same set of elements, a major focus in the research lies
onsimultaneousgraphembedding.Inthissetting,givenasetofgraphswiththesame
r
a vertex-set,thegoalistofindasetofpointsintheplaneandamappingbetweenthese
points and the vertices of the graphs such that placing each vertex on the point it is
mappedtoyieldsaplanarembeddingforeachofthegraphs,iftheyaredisplayedsep-
arately. Problems of this kind frequently arise when dealing with the visualization of
evolvingnetworksandwiththevisualizationofhugeandcomplexrelationships,asin
thecaseofthegraphoftheWeb.
Among the many variants of this problem, the most important and natural one is
the geometric simultaneous embedding. Given two graphs G = (V,E′) and G =
1 2
(V,E′′), the task is to find a set of points P and a bijection M : P → V that induce
planarstraight-lineembeddingsforbothG andG .
1 2
In the seminal paper on this topic [2], Brass et al. proved that geometric simulta-
neous embeddings of pairs of paths, pairs of cycles, and pairs of caterpillars always
exist.Acaterpillarisatreesuchthatdeletingallitsleavesyieldsapath.Ontheother
hand,manynegativeresultshavebeenshown.Brassetal.[2]presentedapairofouter-
planargraphsnotadmittinganysimultaneousembeddingandprovidednegativeresults
for three paths, as well. Erten and Kobourov [4] found a planar graph and a path not
allowing any simultaneous embedding. Geyer et al. [12] proved that there exist two
treesthatdonotadmitanygeometricsimultaneousembedding.However,thetwotrees
used in the counterexample have common edges, and so the problem is still open for
edge-disjointtrees.
Themostimportantopenprobleminthisareaisthequestionwhetheratreeanda
pathalwaysadmitageometricsimultaneousembeddingornot.Inthispaperweanswer
thisquestioninthenegative.
Manyvariantsoftheproblem,wheresomeconstraintsarerelaxed,havebeenstud-
ied in the literature. If the edges do not need to be straight-line segments, a famous
resultofPachandWenger[14]showsthatanynumberofplanargraphsadmitasimul-
taneousembedding,sinceitstatesthatanyplanargraphcanbeplanarlyembeddedon
anygivensetofpointsintheplane.However,thesameresultdoesnotholdiftheedges
thataresharedbytwographshavetoberepresentedbythesameJordancurve.Inthis
settingtheproblemiscalledsimultaneousembeddingwithfixededges[9,11,6].
The research on this problem opened a new exciting field of problems and tech-
niques, like ULP trees and graphs [5,7,8], colored simultaneous embedding [1], near-
simultaneousembedding[10],andmatcheddrawings[3],deeplyrelatedtothegeneral
fundamentalquestionofpoint-setembeddability.
In this paper we study the geometric simultaneous embedding problem of a tree
andapath.Weanswerthequestioninthenegativebyprovidingacounterexample,that
is,atreeand a path notadmitting any geometric simultaneous embedding. Moreover,
sincethetreeandthepathusedinourcounterexampledonotshareanyedge,wealso
negativelyanswerthequestionontwoedge-disjointtrees.
Themainideabehindourcounterexampleistousethepathtoenforceapartofthe
tree to be in a certain configuration which cannot be drawn planar. Namely, we make
useoflevelnonplanar trees[5,8],thatis,treesnotadmittingany planarembedding if
theirverticeshavetobeplacedinsidecertainregionsaccordingtoaparticularleveling.
The tree of the counterexample contains many copies of such trees, while the path is
used to create the regions. To prove that at least one copy has to be in the particular
levelingthatdeterminesacrossing,weneedaquitehugenumberofvertices.However,
suchahugenumberisoftenneededjusttoensuretheexistenceofparticularstructures
playingaroleinourproof.Amuchsmallercounterexamplecouldlikelybeconstructed
withthesametechniques,butwedecidedtopreferthesimplicityoftheargumentations
ratherthanthesearchfortheminimumsize.
The paper is organized as follows. In Sect. 2 we give preliminary definitions and
weintroducetheconceptoflevelnonplanartrees.InSect.3wedescribethetreeT and
the path P used in the counterexample. In Sect. 4 we give an overview of the proof
2
5 7 6 l1 5 7 a1
l
1 1 6
1
l2 a2 1 l2
2 3 4 2 3
l 3
4 3 2 4 a
3
5 8 6 9 7 10 8 10 9 l4 a48 10 9 l3
(a) (b) (c)
Fig.1. (a) A tree T . (b) A level nonplanar tree T whose underlying tree is T . (c) A
u u
region-levelnonplanartreeT whoseunderlyingtreeisT .
u
thatT andP donotadmitanygeometricsimultaneousembedding,whileinSect.5we
givethedetailsofsuchaproof.InSect.6wepresentanalgorithmforthesimultaneous
embeddingofatreeofdepth2andapath,andinSect.7wemakesomefinalremarks.
2 Preliminaries
A (undirected) k-level tree T = (V,E,φ) on n vertices is a tree T′ = (V,E), called
the underlying tree of T, together with a leveling of its vertices given by a function
φ : V 7→ {1,...,k}, such that for every edge (u,v) ∈ E, it holds φ(u) 6= φ(v)
(See[5,8]).AdrawingofT = (V,E,φ)isaleveldrawingifeachvertexv ∈ V such
that φ(v) = i is placed on a horizontal line l = {(x,i) | x ∈ R}. A level drawing
i
ofT isplanarifnotwoedgesintersectexcept,possibly,atcommonend-points.Atree
T =(V,E,φ)islevelnonplanarifitdoesnotadmitanyplanarleveldrawing.
Weextendthisconcepttotheoneofregion-leveldrawingbyenforcingthevertices
ofeachleveltolieinsideacertainregionratherthanonahorizontalline.Letl ,...,l
1 k
be k pairwise non-crossing straight lines and let r ,...,r be the regions of the
1 k+1
planesuchthatanystraight-linesegmentconnectingapointinr andapointinr ,with
i h
1≤i<h≤k+1,cutsallandonlythelinesl ,l ,...,l ,inthisorder.Adrawing
i i+1 h−1
ofak-leveltreeT =(V,E,φ)iscalledregion-leveldrawingifeachvertexv ∈V such
that φ(v) = i is placed inside region r . A region-level drawing of T is planar if no
i
two edges intersect except, possibly, at common end-points. A tree T = (V,E,φ) is
region-levelnonplanarifitdoesnotadmitanyplanarregion-leveldrawing.
The4-leveltreeT whoseunderlyingtreeisshowninFig.1(a)hasbeenshowntobe
levelnonplanar[8](seeFig.1(b)).InthenextlemmaweshowthatT isalsoregion-level
nonplanar(seeFig.1(c)).
Lemma1. The 4-level tree T whose underlying tree is shown in Fig. 1(a) is region-
levelnonplanar.
Proof: Refer to Fig. 1(c). First observe that, in any possible region-level planar
drawing of T, the paths p = v ,v ,v and p = v ,v ,v define a polygon Q (a
1 5 2 8 2 6 3 9 2
3
B
i 3 3
3 3
3 3
3 3
j 2 2 2
h
r 1
(a) (b)
Fig.2. (a) A schematization of the complete tree T . Joints and stabilizers are small
circles, branches are solid triangles, while complete subtrees connected to a joint are
dashedtriangles.(b)AschematizationofabranchB .
i
polygon Q ) inside region r (region r ). We have that v is inside Q , as otherwise
3 2 3 1 2
oneofedges(v ,v )or(v ,v )wouldcrossoneofp orp .Hence,vertexv hasto
1 2 1 3 1 2 4
beinsideQ ,asotherwiseedge(v ,v )wouldcrossoneofp orp .However,inthis
3 1 4 1 2
case,thereisnoplacementforverticesv andv thatavoidsacrossingbetweenone
7 10
ofedges(v ,v )or(v ,v )andoneofthealreadydrawnedges. (cid:3)
4 7 4 10
Lemma 1 will be vital for proving that there exist a tree T and a path P not ad-
mittinganygeometricsimultaneousembedding.Infact,T containsmanycopiesofthe
underlyingtreeofT,whileP connectsverticesofT insuchawaytocreatetheregions
satisfyingtheaboveconditionsandtoenforceatleastoneofsuchcopiestolieinside
theseregionsaccordingtothelevelingmakingitnonplanar.
3 TheCounterexample
InthissectionwedescribeatreeT andapathP notadmittinganygeometricsimul-
taneousembedding.
3.1 TreeT
ThetreeT containsarootrandq verticesj ,...,j atdistance1fromr,calledjoints.
1 q
Each joint j , with h = 1,...,q, is connected to x copies B ,...,B of a subtree,
h 1 x
calledbranch,andtol := (s−1)4·32·xverticesofdegree1,calledstabilizers.See
Fig. 2(a). Each branch B consists of a root r , (s−1)·3 vertices of degree (s−1)
i i
adjacenttor ,and(s−2)·(s−1)·3leavesatdistance2fromr .Verticesbelonging
i i
toabranchB arecalledB-verticesanddenotedby1−,2−,or3−vertices,according
i
totheirdistancefromtheirjoint.Fig.2(b)displays1−,2−,and3−verticesofabranch
B .
i
Because of the huge number of vertices, in the rest of the paper, for the sake of
readability, we use variables n, s, and x as parameters describing the size of certain
4
configurations. Such parameters will be given a value when the technical details of
theargumentations aredescribed.Atthisstagewejustclaimthatatotalnumbern ≤
(cid:0)27·3·x+2(cid:1)ofvertices(seeLemmata5and4)sufficesforthecounterexample.
3
As a first observation we note that, despite the oversized number of vertices, tree
T has limited depth, that is, every vertex is at distance from the root at most 4. This
leadstothefollowingproperty.
Property1. Anypathoftreeedgesstartingattheroothasatmost3bends.
3.2 PathP
Path P is given by describing some basic and recurring subpaths on the vertices of
T and how such subpaths are connected to each other. The idea is to partition the set
of branches B adjacent to each joint j into subsets of s branches each and to con-
i h
necttheirverticeswithpathedges,accordingtosomefeaturesofthetreestructure,so
definingthefirstbuildingblock,calledcell.Then,cellsbelongingtodifferentbranches
areconnectedtoeachother,hencecreatingstructures,calledformations,forwhichwe
can ensure certain properties regarding the intersection between tree and path edges.
Further,differentformationsareconnectedtoeachotherbypathedgesinsuchawayto
createbiggerstructures,calledextendedformations,whichare,intheirturn,connected
tocreateasequenceofextendedformations.
Allofthesestructuresareconstructedinsuchawaythatthereexistsasetofcells
such that any four of its cells, connected to the same joint and being part of the same
formationorextendedformation,containaregion-levelnonplanartreeforanypossible
leveling,wherethelevelscorrespondtocells.Hence,provingthatfourofsuchcellslie
indifferentregionssatisfyingthepropertiesofseparationdescribedaboveisequivalent
to proving the existence of a crossing in the tree. This allows us to consider only the
bigger structures instead of dealing with single copies of the region-level nonplanar
tree.
Inthefollowingwedefinesuchstructuresmoreformallyandstatetheirproperties.
Cell:ThemostbasicstructuredefinedbyP isdefinedbylookingathowitconnects
verticesofsomebranchesB connectedtothesamejointj ofT .Considerasetofs
i h
branchesB ,i = 1,...,s,connectedtoj .Assumetheverticesofalevelinsideeach
i h
treetobearbitrarilyordered.Foreachr =1,...,s,defineacellc (h)tobecomposed
r
ofitshead,itstail,andanumbertof stabilizersofj .
h
Theheadofc (h)consistsoftheunique1-vertexofB ,thefirstthree2-verticesof
r r
eachbranchB ,with1 ≤ k ≤ sandk 6= r,thatarenotalreadyusedinacellc (h),
k a
with1 ≤ a < r,and,foreach2-vertexnotinc (h)andnotinB ,thefirst3-vertices
r r
notalreadyusedinacellc (h),with1≤a<r.
a
Thetailofc (h)consistsofasetof3·s·(s−1)2branchesB adjacenttoj .This
r k h
setispartitionedinto3·(s−1)2subsetsofssubtreeseach.Theverticesofeachofthe
subsetsaredistributedbetweenthecellsinthesamewayasfortheverticesofthehead.
Thisimpliesthateachcellcontainsone1-vertex,3·(s−1)2-vertices,and3·(s−
2)·(s−1) 3-vertices of the head, an additional 3·(s−1)2 1-vertices, 32 ·(s−1)3
2-vertices,and32·(s−2)·(s−1)33-verticesofthetail,plus32·(s−1)4stabilizers.
5
2 2 2 3 3 2 2 3 3
1 1 1 1 1 1
j
h
Fig.3.Acell.B-verticesoftheheadaredepictedbylargewhitecircles,B-verticesof
thetailarelargegreycircles,B-verticesnotpartofthecell(showingthetreestructure)
are small grey circles and stabilizers are small white cirlces. Tree edges are grey and
pathedgesareblack.
Path P inside cell c (h) visits the vertices in the following order: It starts at the
r
unique 1-vertex of the head, then it reaches all the 2-vertices of the head, then all the
3-verticesofthehead,thenallthe2-verticesofthetail,andfinallyallthe3-verticesof
thetail,visitingeachsetinarbitraryorder.Aftereachoccurrenceofa2-or3-vertexof
thehead,P visitsa1-vertexofthetail,andaftereachoccurrenceofa2-ora3-vertex
ofthetail,itvisitsastabilizerofjointj (seeFig.3).
h
Note that, by this construction, for each joint there exists a set of cells such that
eachsubsetofsizefourcontainsregion-levelnonplanartreeswithallpossiblelevelings,
wherethelevelscorrespondtothemembershipoftheverticestoacell.Wenowdefine
two bigger structures describing how cells of this set are connected to cells of sets
connectedtootherjoints.
Formation:Inthedefinitionofacellwedescribedhowthepathtraversesthrough
onesetofbranchesconnectedtothesamejoint.Nowwedescribehowcellsfromfour
differentsetsareconnected.
AformationF(H),H =(h ,h ,h ,h )consistsof592cells,namelyof148cells
1 2 3 4
c (h ) from the set of cells constructed above for each 1 ≤ i ≤ 4. Path P connects
r i
these cells in the order ((h h h )37h37)4, that is, P repeats four times the following
1 2 3 4
sequence: It connects c (h ) to c (h ), then to c (h ), then to c (h ), and so on till
1 1 1 2 1 3 2 1
c (h ), from which it then connects to c (h ), to c (h ), and so on till c (h ) (see
37 3 1 4 2 4 37 4
Fig.4(a)).Aconnectionbetweentwoconsecutivecellsc (a)andc (b)isdonewithan
r r
edgeconnectingtheendverticesofthepartsP (c (a))andP (c (b))ofP restrictedto
r r
theverticesofc (a)andc (b),respectively.Namely,theuniquevertexinc (a)having
r r r
degree1bothinP (c (a))andinT isconnectedtotheuniquevertexinc (b)having
r r
degree1inP (c (b))butnotinT .Thefollowingpropertyholds:
r
Property2. ForanyformationF(H)andanyjointj ,withh∈H,iffourcellsc (h)∈
h r
F(H)arepairwiseseparatedbystraightlines,thenthereexistsacrossinginT .
ExtendedFormation:Formationsareconnectedbythepathinaspecialsequence,
definedasextendedformationanddenotedbyEF(H),whereH =(H =(h ,...,h ),
1 1 4
H = (h ,...,h ),...,H = (h ,...h ))isatupleof4−tuplesofdisjointin-
2 5 8 x 4x−3 4x
dexes of joints (see Fig. 4(b)). Let F1(Hi),...,Fy−xy(Hi) be y − xy formations not
6
c (j ) c (j )
c37(j 1 ) 37 3 37 4
c(j )
2 1
c(j ) c(j )
jh1 1 1 jh2 jh3 jh41 4
r
(a) (b)
Fig.4.(a)Aformation.Treeedgesaredepictedbygreyandpathedgesbyblacklines.
Please note in this figure also the bundle of tree edges connecting the different cells
belonging to the same branch. (b) A subsequence (H ,...,H )2 of an extended for-
1 x
mation. Formations are inside a table to represent the 4-tuple they belong to and to
emphasizethatineachrepetition(arowofthetable)aformationatacertain4-tupleis
missing.
belonging to any other extended formation and composed of cells of the same set S.
These formations are connected in the order (H ,H ,...,H )y, but in each of these
1 2 x
y repetitionsoneH ismissing.Namely,inthek-threpetitionthepathdoesnotreach
i
anyformationatH ,withm=k mod x.Wesaythatthek-threpetitionhasadefect
m
atm.Wecallasubsequence(H ,H ,...,H )x afullrepetitioninsideEF(J).Afull
1 2 x
repetitionhasexactlyonedefectateachtuple.
Notethatthesizeofscannowbefixedasthenumberofformationscreatingrep-
etitions inside one extended formation times the number of cells inside each of these
formations,thatiss:=(y−y)·37·4.Weclaimthatx≤7·32·223andy ≤72·33·226
x
is sufficient throughout the proofs. However, for readability reasons, we will keep on
usingvariablesxandyintheremainderofthepaper.
SequenceofExtendedFormation:Extendedformationsareconnectedbythepath
inaspecialsequence,calledsequenceofextendedformationsanddenotedbySEF(H),
where H = (H∗,...,H∗ ) is a 12−tuple of tuples of 4−tuples. For each tuple H∗,
1 12 i
where i = 1,...,12, consider 110 extended formations (EF (H∗),...,EF (H∗ )),
i 1 i 12
withi = 1,...,110, notalready belonging toany other sequence ofextended forma-
tions.TheseextendedformationsareconnectedinsideSEF(H)intheorder(H∗,...,
1
H∗ )(120).Thereexisttwotypesofsequencesofextendedformations.Namely,inthe
12
firsttypethereisoneextendedformationmissingineachsubsequence(H∗,...,H∗ ),
1 12
thatwecalldefect,asfortheextendedformations.Inthesecondtype,twoconsecutive
extendedformationsaremissing.Namely,inthek-threpetitionthepathskipstheex-
tendedformationsconnectingatH∗ andatH∗ ,withm = kmod12.Inthiscase,
m m+1
wesaythattherepetitionhasadoubledefect.
Since, for each set of 48x joints, (48x)! different disjoint sequences of extended
formationsexist,wejustconsiderthesequenceswheretheorderdefinedbythetupleis
theorderofthejointsaroundtheroot.
7
c (h)
1 j
e1 e2 j jh1’ h’2
h
c’(h’) c’(h’) 1
c (h) j
h
j 2 j 2
h h’
r r
(a) (b)
Fig.5. (a) A passage between cells c (h), c (h), and c′(h′). (b) Two interconnected
1 2
passages.
4 Overview
Inthissectionwepresentthemainargumentationsleadingtothefinalconclusionthat
thetreeT andthepathP describedinSect.3donotadmitanygeometricsimultaneous
embedding.Themainideainthisproofschemeistousethestructuresgivenbythepath
tofixapartofthetreeinaspecificshapecreatingspecificrestrictionsfortheplacement
ofthefurthersubstructuresofT andofP attachedtoit.
Wefirstgivesomefurtherdefinitionsandbasictopologicalpropertiesontheinter-
action among cells that are enforced by the preliminary arguments about region-level
planardrawingsandbytheorderinwhichthesubtreesareconnectedinsideoneforma-
tion.
Passage:Considertwocellsc (h),c (h)thatcannotbeseparatedbyastraightline
1 2
andacellc′(h′),withh′ 6=h.WesaythatthereexistsapassageP betweenc ,c ,and
1 2
c′ifthepolylinegivenbythepathofc′separatesverticesofc fromverticesofc (see
1 2
Fig.5(a)).Sincethepolylinecannotbestraight,thereisavertexofc′ lyinginsidethe
convexhulloftheverticesofc ∪c ,whichimpliesthefollowing.
1 2
Property3. Inapassagebetweencellsc ,c ,andc′thereexistatleasttwopath-edges
1 2
e ,e ofc′suchthatbothe ande areintersectedbytree-edgesconnectingverticesof
1 2 1 2
c toverticesofc .
1 2
FortwopassagesP betweenc (h ),c (h ),andc′(h′),andP betweenc (h ),
1 1 1 2 1 1 2 3 2
c (h ),andc′(h′)(w.l.o.g.,weassumeh < h′,h < h′,andh < h ),wedistin-
4 2 2 1 1 2 2 1 2
guishthreedifferentconfigurations:(i)Ifh′ < h ,P andP areindependent;(ii)if
1 2 1 2
h′ <h′,P isnestedintoP ;and(iii)ifh <h′ <h′,P andP areinterconnected
2 1 2 1 2 1 2 1 2
(seeFig.5(b)).
Doors:Letc (h),c (h),andc′(h′)bethreecellscreatingapassage.Considerany
1 2
triangle given by a vertex v′ of c′ inside the convex hull of c ∪c and by any two
1 2
vertices of c ∪ c . This triangle is a door if it encloses neither any other vertex of
1 2
c1,c2 nor any vertex of c′ thatis closer than v′ to jh′ in T .Adoor isopen if no tree
edgeincidenttov′crossestheoppositesideofthetriangle,thatis,thesidebetweenthe
verticesofc andc (seeFig.6(a)),otherwiseitisclosed(seeFig.6(b)).
1 2
8
Considertwojointsj andj ,withh,a,h′,bappearinginthiscircularorderaround
a b
theroot.Anypolylineconnectingtheroottoj ,thentoj ,andagaintotheroot,without
a b
crossingtreeedges,musttraverseeachdoorbycrossingboththesidesadjacenttov′.
If a door is closed, such a polyline has to bend after crossing one side adjacent to v′
andbeforecrossingtheotherone.Also,iftwopassagesP andP areinterconnected,
1 2
eitherallthecloseddoorsofP aretraversedbyapathoftree-edgesbelongingtoP
1 2
orallthecloseddoorsofP aretraversedbyapathoftree-edgesbelongingtoP (see
2 1
Fig.5(b)).
In the rest of the argumentation we will exploit the fact that the closed door of a
passage requests a bend in the tree to obtain the claimed property that a large part of
T hastofollowthesameshape.Inviewofthis,westatethefollowinglemmatarelating
theconceptsofdoors,passages,andformations.
Lemma2. ForeachformationF(H),withH = (h ,...,h ),thereexistsapassage
1 4
betweensomecellsc (h ),c (h ),c′(h )∈F(H),with1≤a,b≤4.
1 a 2 a b
Lemma3. Eachpassagecontainsatleastonecloseddoor.
Fromthepreviouslemmataweconcludethateachformationcontainsatleastone
closeddoor.Toprovethattheeffectsofcloseddoorsbelongingtodifferentformations
canbecombinedtoobtainmorerestrictionsonthewayinwhichthetreehastobend,
weexploitacombinatorialargumentbasedontheRamseyTheorem[13]andstatethat
thereexistsasetofjointspairwisecreatingpassages.
Lemma4. Given a set of joints J = {j1,...,jy}, with |J| = y := (cid:0)27·33·x+2(cid:1), there
existsasubsetJ′ = {j′,...,j′},with|J′| = r ≥ 27·3·x,suchthatforeachpairof
1 r
jointsj′,j′ ∈J′thereexisttwocellsc (i),c (i)creatingapassagewithacellc′(h).
i h 1 2
Now we formally define the claimed property that part of the tree has to follow a
fixed shape by considering how the drawing of the subtrees attached to two different
jointsforcethedrawingofthesubtreesattachedtothejointsbetweenthemintheorder
aroundtheroot.
Enclosingbendpoints:Considertwopathsp ={u ,v ,w }andp ={u ,v ,w }.
1 1 1 1 2 2 2 2
The bendpoint v of p encloses the bendpoint v of p if v is internal to triangle
1 1 2 2 2
△(u ,v ,w ).SeeFig.7(a).
1 1 1
b b
y y
b b
b j’ b j’
j j
b
b x
x
r r
(a) (b)
Fig.6.(a)Anopendoor.(B)Acloseddoor.
9
v
1
v
u 2 w
1 1
cs
cs
cs 3
1 cs 4
u w 2
2 2
(a) (b)
Fig.7.(a)Anenclosingbendpoint.(b)A3-channelanditschannelsegments.
Channels: Consider a set of joints J = {j ,...,j } in clockwise order around
1 k
the root. The channel c of a joint j , with i = 2,...,k −1, is the region given by
i i
thepairofpaths,onepathofj andonepathofj ,withthemaximumnumberof
i−1 i+1
enclosingbendpointswitheachother.Wesaythatc isanx-channelifthenumberof
i
enclosingbendpointsisx.Observethat,byProperty1,x≤3.A3-channelisdepicted
inFig.7(b).Notethat,givenanx-channelc ofj ,alltheverticesofthesubtreerooted
i i
atj thatareatdistanceatmostxfromtherootlieinsidec .
i i
Channel segments: An x-channel c is composed of x+1 parts called channel
i
segments (see Fig. 7(b)). The first channel segment cs is the part of c that is visible
1 i
fromtheroot.Theh-thchannelsegmentcs istheregionofc disjointfromcs that
h i h−1
isboundedbytheelongationsofthepathsofj andj aftertheh-thbend.
i−1 i+1
Observe that, sincethe channels arecreated by tree-edges, any tree-edge connect-
ing vertices in the channel has to be drawn inside the channel, while path-edges can
cross other channels. In the following we study the relationships between path-edges
andchannels.Thefollowingpropertydescendsfromthefactthateverysecondvertex
reachedbyP inacelliseithera1-vertexorastabilizer.
Property4. Foranypathedgee=(a,b),atleastoneofaandblieinsideeithercs or
1
cs .
2
Blocking cuts:Ablocking cut isapath edge connecting twoconsecutive channel
segmentsbycuttingsomeoftheotherchannelstwice.SeeFig.8.
Property5. Let c be a channel that is cut twice by a blocking cut. If c has vertices in
boththechannelsegmentscutbythepathedge,thenithassomeverticesinadifferent
channelsegment.
Proof:Considertheverticeslyinginthetwochannelsegmentsofc.Inordertoconnect
theminT ,avertexv isneededinthebendpointareaofc.However,inordertohave
pathconnectivitybetweenvandtheverticesinthetwochannelsegments,somevertices
inadifferentchannelsegmentareneeded. (cid:3)
InthefollowinglemmaweshowthatinasetofjointsasinLemma4itispossible
to find a suitable subset such that each pair of paths of tree-edges starting from the
rootandcontainingsuchjointshasatleasttwocommonenclosingbendpoints,which
impliesthatmostofthemcreate2-channels.
10