O conteudo do pre s ente relator io e deunica re sp onsabilidade dos autore s

The contents of thi s rep ort are the sole re sp ons ibility of the authors

Ear decomp o s itions of matching covere d graphs

Marcelo H Carvalho

UFMS Campo Grande Brasil

Claudio L Lucchesi

UNICAMP Campinas Brasil

USRMurty

University of Waterloo Waterloo Canada

Relator io Tecnico IC

Fevere iro de



Ear decomp o s itions of matching covere d graphs

z y

Claudio L Lucche s i Marcelo H Carvalho

UNICAMP Campinas Bras il UFMS Camp o Grande Bras il

x

USRMurty

Univers ity of Waterlo o Waterlo o Canada

February

Ab stract

Ear decompositions of matching covered graphs are important for understanding their

structure By exploiting the properties of the dependence relation introduced by Carvalho

and Lucchesi in we are able to provide simple proofs of several wel lknown theorems

concerning ear decompositions Our method actual ly provides proofs of generalizations

of these theorems For example we show that every matching covered graph G dierent

from K has at least  edgedisjoint removable ears where  is the maximum

2

of G This shows that any matching covered graph G has at least ! dierent ear

decompositions and thus is a generalization of the fundamental theorem of Lovasz and

Plummer establishing the existence of ear decompositions We also show that every

C has  2 edges each of which is a removable edge brick G dierent from K and

6 4

in G that is an edge whose deletion from G results in a matching covered graph This

generalizes a wel lknown theorem of Lovasz Using this theorem we give a simple proof

of another theorem due to Lovasz which says that every nonbipartite matching covered

graph has a canonical ear decomposition that is one in which either the third graph in

the sequence is an oddsubdivision of K or the fourth graph in the sequence is an odd

4

subdivision of C Our method in fact shows that every nonbipartite matching covered

6

graph has a canonical ear decomposition which is optimal that is one which has as few

double ears as possible Most of these results appear in the PhDthesis of the rst

author written under the supervision of the second author

Intro duction

A matching covere d graph i s a nontr ivial connecte d graph in which every e dge i s in some

p erfect matching There i s an extens ive literature on matching covere d graphs For a hi story

of the sub ject and for all terminology and notation not dene d here we refer the reader to

Lovasz and Plummer Lovasz or Murty

AMS sub ject class ication C

y

Supp orte d by UFMS and CAPES Bras il

z

Supp orte d by grants f rom CNPq and FAPESP

x

Work partially done dur ing thi s authors vi s it to UNICAMP Bras il f rom Septemb er to Decemb er

Carvalho Luccche s i and Murty

Two typ e s of decomp os itions of matching covere d graphs namely tight cut decomp os i

tions and ear decomp os itions have playe d s ignicant role s in the development of the sub ject

For example b oth the s e decomp os itions are e ss ential ingre dients in Lovaszs chracter iza

tion of the matching lattice Here we us e tight cut decomp os itions in our study of ear

decomp os itions For all the relevant denitions and theorems concer ning the tight cut de

comp os ition of a matching covere d graph we refer the reader to the source s cite d ab ove We

recall b elow some of the bas ic denitions relate d to ear decomp os itions

Let G b e a subgraph of a graph G and let P b e a path in G E G A s ingle e ar of

G i s a path P of o dd length in G such that i b oth the ends of P are in V G and ii P

i s inter nally di sjoint f rom G The following theorem provide s a decomp os ition of bipartite

matching covere d graphs

Theorem Given any bipartite matching covered graph G there exists a sequence

G G G G

1 2 r

of matching covered subgraphs of G where i G K and ii for i r G is

1 2 i+1

the union of G and a single ear of G

i i

The s equence G G G of subgraphs of G with the ab ove prop ertie s i s an ear de

1 2 r

comp os ition of G in which each memb er of the s equence i s a matching covere d subgraph of G

Such decomp os itions do not exi st for nonbipartite matching covere d graphs For example

K has no ear decomp os ition as in Theorem However every matching covere d graph

4

has an ear decomp os ition with a slight relaxation of the ab ove denition For de scr ibing that

decomp os ition we require the notion of a double ear

A double e ar of a matching covere d graph G i s a pair of vertex di sjoint s ingle ears of

G An e ar of G i s e ither s ingle ear or double ear of G

An e ar decomp o s ition of a matching covere d graph G i s a s equence

G G G G

1 2 r

of matching covere d subgraphs of G where i G K and ii for i r G i s

1 2 i+1

the union of G and an ear s ingle or double of G The following fundamental theorem was

i i

e stabli she d by Lovasz and Plummer See

Theorem The twoe ar Theorem Every matching covered graph has an ear decom

position

There are var ious pro ofs of thi s theorem The or iginal pro of due to Lovasz and Plummer

i s somewhat involve d Later on Little and Rendl gave a s impler pro of Recently another

s imple pro of was given by Szigeti

A matching covere d subgraph H of a graph G i s nice if every p erfect matching of H

extends to a p erfect matching of G It i s easy to s ee that all the subgraphs in an ear

decomp os ition of a matching covere d graph must b e nice matching covere d subgraphs of the

graph In all the previous approache s an ear decomp os ition of a matching covere d graph

Ear decomp os itions of matching covere d graphs

G i s e stabli she d by showing how a nice matching covere d prop er subgraph G of G can b e

i

extende d to a nice matching covere d subgraph G by the addition of a s ingle or double ear

i+1

Our approach i s to build an ear decomp os ition of a matching covere d graph in the revers e

order To state our theorem preci s ely we nee d to dene the notion of a removable ear in a

matching covere d graph

Let G b e a matching covere d graph and let P b e a path in G Then P i s said to b e a

removable s ingle e ar in G if i P i s an o dd path which i s inter nally di sjoint f rom the re st

of the graph and the graph obtaine d f rom G by deleting all the e dge s and inter nal vertice s of

P i s matching covere d A removable ear of length one i s a removable e dge The notion

of a removable double e ar in G i s s imilarly dene d A removable e ar in G i s e ither a

s ingle or double ear which i s removable

In trying to e stabli sh the exi stence of ear decomp os itions with sp ecial prop ertie s it i s often

convenient to nd the subgraphs in the ear decomp os ition in the revers e order starting with

G G Thus after obtaining a subgraph G in the s equence which i s dierent f rom K we

r i 2

nd a suitable removable ear s ingle or double and obtain G f rom G by removing that ear

i1 i

f rom G For example to show that a matching covere d graph G has an ear decomp os ition

i

it suce s to show that every matching covere d graph dierent f rom K has a removable ear

2

Thi s i s our approach We in f act prove the folowing more general theorem

Theorem Any matching covered graph G dierent from K has removable ears single

2

or double

Clearly Theorem i s a sp ecial cas e of Theorem In f act shows that every

matching covere d graph G has at least dierent ear decomp os itions

Supp os e that

G G G G

1 2 r

i s an ear decomp os ition of a matching covere d graph G If a memb er G of the s equence i s

i

obtaine d f rom its pre dece ssor G by adding a s ingle double ear then we shall say that

i1

G i s obtaine d f rom G by the addition of a s ingle double ear In any ear decomp os ition

i i1

of a nonbipartite matching covere d graph G there must b e at least one double ear addition

Furthermore if G i s the rst memb er of an ear decomp os ition which i s obtaine d by a double

i

1

ear addition then G G are bipartite and G G G are nonbipartite

1 i1 i r

In any ear decomp os ition by denition G i s K and G i s an even circuit It i s not

1 2 2

dicult to check that if G i s nonbipartite then it must in f act b e an o dd sub divi s ion of

3

K and if G i s bipartite and G i s nonbipartite then G must b e an o dd sub divi s ion of

4 3 4 4

C We shall refer an ear decomp os ition

6

G G G G

1 2 r

of a nonbipartite matching covere d graph G as a canonical e ar decomp o s ition if e ither

its third memb er G or its fourth memb er G i s nonbipartite The following fundamental

3 4

theorem was prove d by Lovasz in

1

We only cons ider ear decomp os ition s which are ne By thi s we mean that a double ear P P i s adde d

1 2

to G to obtain G only if ne ither G P nor G P i s matching covere d

i i+1 i 1 i 2

Carvalho Luccche s i and Murty

Theorem The canonical e ar decomp o s ition Theorem Every nonbipartite match

ing covered graph G has a canonical ear decomposition

In Lovasz gave s everal applications of the ab ove theorem to extremal problems in

Later on he us e d it to de duce the following theorem which was crucial for hi s

work on the matching lattice

C Theorem The removable e dge Theorem Every brick dierent from K and

6 4

has a removable edge

Lovaszs or iginal pro of of Theorem was quite involve d A s impler pro of was given

by Carvalho and Lucche s i Our approach here shall b e to prove Theorem rst and

then to der ive f rom it We shall in f act prove the following generalization of

Theorem Every brick dierent from K and C has at least removable edges

4 6

A pro of of the ab ove theorem was rst given by Carvalho and Lucche s i in The pro of

we give here i s along the same line s but i s somewhat s impler

Any two ear decomp os itions of a bipartite matching covere d graph have the same length

However the length of an ear decomp os ition of a nonbipartite matching covere d graph de

p ends on the numb er of double ears us e d in the decomp os ition An optimal ear decomp os i

tion of a matching covere d graph G i s one which us e s as few double ears as p oss ible Optimal

ear decomp os ition of G have b een us e d by Carvalho and Lucche s i for determining a bas i s

of the matching lattice of G In thi s connection they prove d the following generalization of

Theorem Optimal canonical e ar decomp o s ition Theorem Every nonbipartite

matching covered graph G has an optimal ear decomposition which is canonical

One mer it of our approach i s that we are able to prove the ab ove theorem f rom

directly by a s imple inductive argument

In the next s ection we recall the denition of the dep endence relation that we allude d

to in the abstract and state some of its salient prop ertie s In s ection we give a pro of of

Theorem In s ection we give a pro of of Theorem In s ection we give pro ofs

of Theorems and

A dep endence relation

Let G b e a matching covere d graph and let e and f b e any two e dge s of G Then e dep ends

on f or e implie s f if every p erfect matching that contains e also contains f Equivalently

e dep ends on f if e i s not admi ss ible in G f

We shall wr ite e f to indicate that e dep ends on f Clearly i s reexive and

trans itive It i s convenient to vi sualize in terms of the digraph it dene s on the s et of

e dge s of G See Figure for an illustration A study of thi s relation as shown by its many

Ear decomp os itions of matching covere d graphs

c

1 c

a

1

a

e

c c

c

c

1

c

1 c

1

b b

a

a

1

a

a

a

1 1

a

d

d

b b

e

Figure

imp ortant us e s in Carvalhos the s i s tur ns out to b e very us eful for understanding the

structure of matching covere d graphs

The folowing lemmas can b e prove d eas ily us ing Tuttes theorem

Lemma See Let G be a graph which has a Then an edge e of

G is admissible if and only if there is no barrier which contains both the ends of e

Lemma Let G be a matching covered graph and let e and f be any two edges of G

Then the relation e f holds if and only if there exists a barrier B of G f such that i

both the ends of e are in B and ii the ends of f lie in dierent components of G f B

Pro of If G f has a barr ier B sati sfying the ab ove prop ertie s then clearly e f Supp os e

now that e f or that e i s not admi ss ible in G f Clearly G f has a p erfect matching

Therefore by G f has a barr ier B such that b oth ends of e are in B On the other

hand s ince e i s admi ss ible in G B cannot b e a barr ier in G its elf It follows that the ends

of f lie in dierent comp onents of G f B

We shall say that two e dge s e and f are mutually dep endent if e f and f e In

thi s cas e we shall wr ite e f Clearly i s an equivalence relation on E G By identifying

the vertice s in the equivalence class e s in the digraph repre s enting dep endence relation

on E G we obtain the digraph D G repre s enting the dep endence relation on the s et

of equivalence class e s Thi s digraph i s clearly acyclic The source s in thi s digraph are calle d

minimal classes If e i s an e dge then a source Q in the comp onent of D G containing e or

more preci s ely containing the equivalence class containing e i s said to b e a minimal clas s

induce d by e The following lemma i s an imme diate cons equence of thi s denition

Carvalho Luccche s i and Murty

Lemma If Q is a minimal class then every edge not in Q is admissible in G Q

Thus if G Q is connected then G Q is matching covered

The equivalence class e s in a br ick have some attractive prop ertie s

Lemma Let G be a brick and let e and f be two edges of G such that e f Then

G e f is bipartite Moreover both ends of e lie in one part of the bipartition and both

ends of f lie in the other part of the bipartition

Pro of By Lemma G f has a barr ier B such that b oth ends of e are in B and the

two ends of f are in dierent comp onents of G f B

Supp os e that there exi sts another e dge e which has b oth its ends in B Let M b e a

p erfect matching in G which contains the e dge e Simple counting shows that f M and

e M But thi s contradicts the hyp othe s i s that e f Therefore e i s the only e dge with

b oth ends in B

Supp os e now that G e f i s not bipartite Then some comp onent H of G e B

i s nontr ivial As G i s a br ick rV H i s not a tight cut in G Hence there exi sts a

p erfect matching M of G such that jM rV H j A s imple counting argument shows

jM rV H j f M and e M Again thi s contradicts the hyp othe s i s Therefore

G e f i s bipartite

Convers ely if G e f i s bipartite then clearly e f

Lemma Let G be a brick and let Q be an equivalence class of G Then jQj

Moreover if jQj then G Q is bipartite

If p oss ible let e f and g b e any three e dge s in Q Then by Lemma E n fe f g Pro of

and E n fe g g are cob oundar ie s of G The symmetr ic dierence of the s e two cob oundar ie s

i s ff g g But the symmetr ic dierence of any two cob oundar ie s of a graph i s a cob oundary

of the graph Thus ff g g i s a cob oundary of G Thi s i s a contradiction b ecaus e G i s

connecte d

In a general matching covere d graph the equivalence class e s can b e arbitrar ily large

However the following re sult can b e prove d us ing an argument s imilar to the one us e d in the

pro of of

Lemma CarvalhoLucche s i Let G be a matching covered graph and let Q be a min

imal class of G If Q does not include an edge cut of size then jQj In particular if

G is a edgeconnected then jQj for any minimal class Q of G

The pro of given by Carvalho and Lucche s i of Theorem in i s bas e d on the ab ove

lemma

Ear decomp os itions of matching covere d graphs

Removable e ars in matching covere d graphs

We shall rst cons ider the cas e s in which G i s e ither a br ick or i s a brace on at least s ix

vertice s We shall then b e able to prove the theorem in general by induction us ing the tight

cut decomp os ition pro ce dure

Lemma Let G be a brick Then G has removable ears

Pro of Let v b e a vertex of degree of G and let e e b e the e dge s incident with v

1 

Let Q Q b e minimal equivalence class e s induce d by e e re sp ectively It follows

1  1 

f rom the denitions that for each i i there exi sts a p erfect matching M such that

i

M contains the e dge s in Q and the e dge e b ecaus e Q b e ing induce d by e dep ends on

i i i i i

e It follows that the M are di stinct and hence the Q are di stinct But by G Q

i i i i

i s matching covere d for each i and by jQ j for i

i

Lemma Let H be a brace on six or more vertices Then every edge e of H is removable

Pro of Let H b e a brace with bipartion U W where jU j jW j By denition for

any subs et X of U or of W jX j jU j we have jN X j jX j

Let e b e any e dge of H Then jN X j jX j for each subs et X of U or

H e

of W jX j jU j Supp os e now that there exi sts a subs et X of U or of W

jX j jU j such that jN X j jX j As H e has a p erfect matching we must in

H e

f act have jN X j jX j Then there exi sts a vertex w W such that jN w j

H e H e

Cons equently jN w j Thi s i s not p oss ible b ecaus e H i s a brace

H

Therefore jN X j jX j for every prop er subs et X of U Hence H e i s a

H e

matching covere d graph

Pro of of Theorem It i s easy to ver ify the statement of the theorem if jV Gj

Let G b e a matching covere d graph on s ix or more vertice s If G i s e ither a br ick or a brace

then the validity of the theorem follows f rom Lemmas and To prove the theorem

in general assume inductively that any matching covere d graph G with jE G jV G j

jE G jV Gj has G e dgedi sjoint removable ears We shall rst deal with a few s imple

cas e s

First cons ider the cas e in which G has two parallel e dge s say e and f Then G f i s

matching covere d and has fewer e dge s than G By induction G f has at least G f e dge

di sjoint removable ears say Q Q Q If e i s not in any one of the s e G f ears

1 2

(Gf )

then they are removable ears in G as well and in addition feg and ff g are also removable

in G On the other hand if e i s one of the Q say e Q then feg ff g Q Q

i 1 2

(Gf )

are removable ears in G Since G f G it follows that G has at least G

e dgedi sjoint removable ears Thus we may assume that G i s s imple

Now cons ider the cas e in which G has two adjacent vertice s say v and w of degree two

Let u b e the ne ighb our of v dierent f rom w and t b e the ne ighb our of w dierent f rom v If

u i s the same vertex as t then it would b e a cut vertex of G which i s imp oss ible Therefore

u and v are di stinct and P u v w t i s a path of length three which i s inter nally di sjoint

Carvalho Luccche s i and Murty

f rom the re st of the graph Obtain the graph G f rom G by deleting v and w and joining u

and t by a new e dge e Clearly G has fewer vertice s than G and G G Therefore

by induction G has removable ears Any removable ear of G that do e s not include the

e dge e i s also a removable ear of G And if Q i s a removable ear of G which include s e

we can obtain a removable ear Q of G f rom Q b e replacing e by the path P It follows that

G has at least G e dgedi sjoint removable ears Thus we may assume that no two of its

vertice s of degree two are adjacent

We divide the re st of the pro of into two main cas e s dep ending on whether or not G i s

bicr itical If G i s bicr itical and i s not a br ick then G has a s eparation cut Thi s i s the

rst cas e we cons ider

Cas e G i s bicr itical and fv v g i s a s eparation of G Firstly note that if there i s

1 2

an e dge f joining v and v in G then G f i s also bicr itical In particular f i s removable in

1 2

G Moreover if Q i s a removable ear of G f then Q i s also a removable ear of G Thus if

there i s an e dge joining v and v we can delete it and apply induction to obtain the require d

1 2

re sult So we may assume that there i s no such e dge in G

Expre ss G in the usual manner as the union of two e dgedi sjoint graphs H H with

1 2

V H V H fv v g Let e v v b e a new e dge Then G H e and G H e

1 2 1 2 1 2 1 1 2 2

are b oth matching covere d Furthermore G G and thus ne ither G nor G i s

1 2 1 2

e ither K

2

Let v b e a vertex of degree in G Without loss of generality we may assume e ither that

v V G n fv v g or that v v Let us rst supp os e that v V G n fv v g Then by

1 1 2 1 1 1 2

induction G has at least e dgedi sjoint removable ears and at most one of them contains

1

the e dge e and G has at least three e dgedi sjoint removable ears and at most one of them

2

contains the e dge e Thus G has at least e dgedi sjoint removable ears

Now supp os e that v v and let and b e the degree s of v in H and H re sp ectively

1 1 2 1 1 2

Then the degree s of v in G and G re sp ectively are and By induction G

1 1 2 1 2 1

has at least e dgedi sjoint removable ears and at most one of them contains the e dge

1

e and G has at least e dgedi sjoint removable ears and at most one of them contains

2 2

the e dge e Thus G has at least e dgedi sjoint removable ears

1 2

To complete the pro of we must cons ider the cas e in which G i s not bicr itical In thi s

cas e G has at least one nontr ivial barr ier and hence has nontr ivial tight cuts which are

barr ier cuts We shall apply the induction hyp othe s i s to graphs obtaine d by contracting the

shore s of suitably chos en barr ier cuts In thi s part of the pro of we shall nd it convenient

to us e the notation de scr ib e d b elow and the lemma that follows

If C rS i s a tight cut of G then we shall denote the graph obtaine d by shr inking S

to a s ingle vertex s by by GfS sg if the name of the vertex s i s irrelevant we shall s imply

S g are the two C contractions of G Similarly if S wr ite GfS g Thus GfS g and Gf

1

S S are k di sjoint subs ets of the vertex s et of G then we shall denote the graph obtaine d

2 k

f rom G by shr inking each of the subs ets S S S to s ingle vertice s by GfS S S g

1 1 k 1 2 k

Ear decomp os itions of matching covere d graphs

Lemma Let G be a matching covered graph Let C rS be a nontrivial tight cut

of G and let Q be a removable ear of Gf S g If either Q is edgedisjoint from C or if

E Q C feg and e is removable in GfS g then Q is also removable in G

With the aid of thi s lemma we can now deal of the remaining cas e in which G has a

nontr ivial barr ier It i s convenient to rst cons ider the cas e in which G has a barr ier of s ize

two Note that if G has a vertex u of degree two with v and w as its ne ighb ours then fv w g

i s a barr ier of s ize two of G Furthermore s ince G do e s not have adjacent vertice s of degree

two ne ither v nor w has degree two Thus if G has a barr ier of s ize two then it has one in

which ne ither vertex has degree two in G

Cas e G has barr iers of s ize two In thi s cas e as note d ab ove there exi sts a barr ier

of s ize two in which ne ither vertex has degree two Let fv v g b e such a barr ier and let S

1 2 1

and S b e the vertex s ets of the two comp onents of G n fv v g

2 1 2

Supp os e one of S S say S i s a s ingleton Then there are at least two e dge s f rom

1 2 2

each of v and v to S So all e dge s in rS are multiple e dge s in GfS g and hence are

1 2 1 1 1

S g i s dierent f rom K and has a vertex whos e degree removable in GfS g Furthermore Gf

1 2 1

S g i s at least G By induction hyp othe s i s GfS g has at least removable ears in Gf

1 1

By Lemma they are also removable in G

In general for i j if there i s more than one e dge f rom v to S then all the s e e dge s

i j

would b e multiple e dge s in GfS g and hence would b e removable e dge s of GfS g Thus

j j

s ince b oth v and v have degree at least three the numb er of e dge s in the cut rS which

1 2 1

are not removable in GfS g plus the numb er of e dge s of rS which are not removable in

1 2

GfS g i s at most two By cons ider ing the cuts rS rS us ing the induction hyp othe s i s

2 1 2

and applying Lemma twice we can de duce that G has at least removable ears

Cas e G i s not bicr itical and the s ize of a smalle st nontr ivial barr ier in

G i s at le ast three Let B b e a minimal nontr ivial barr ier of G Let S S S

1 2 b

where b jB j b e the vertex s ets of the o dd comp onents of G B The minimality of B

implie s that the bipartite graph H GfS S S g obtaine d by shr inking S S S

1 2 k 1 2 b

to s ingle vertice s i s a brace Furthermore s ince jB j H i s a brace on s ix or more vertice s

Therefore by Theorem every e dge of H i s removable

S g GfS g GfS g It i s easy to s ee that none of the s e graphs Cons ider the graphs Gf

1 2 b

can b e K

2

Let v b e a vertex of degree in G Without loss of generality v i s e ither in B or in S

1

First supp os e that v i s in B and let b e the numb ers of e dge s f rom v to S S

1 2 b 1 2

S re sp ectively Then clearly Gf S g GfS g GfS g

b 1 1 2 2 b b

By induction for i k Gf S g has at least e dgedi sjoint removable ears Since

i i

each e dge of rS i s removable in H it follows that for i b G has at least

i i

P

b

S g Thus G has at least e dgedi sjoint removable ears containe d in E Gf

i i

i=1

e dgedi sjoint removable ears

Now cons ider the cas e in which v i s in S Then Gf S g G By induction G

1 1 1

has at least G e dgedi sjoint removable ears Since each e dge of rS i s removable in H 1

Carvalho Luccche s i and Murty

S g are also removable in G it follows that all removable e dge s of Gf

1

Remark The ab ove theorem implie s that every matching covere d graph has at least

removable ears Thi s b ound i s b e st p oss ible b ecaus e the graph on two vertice s and parallel

e dge s joining the two vertice s has exactly ear decomp os itions

Removable e dge s in br icks

The following lemma e stabli she s a prop erty of equivalent e dge s in bipartite matching covere d

graphs It i s us e d in the pro of of Theorem

Lemma Let H be a bipartite matching covered graph and let e and f be two distinct

equivalent edges of H Then H fe f g is disconnected

By there i s a barr ier in H f which contains b oth the ends of the e dge e Cho os e Pro of

a maximal barr ier B in H f which contains b oth the ends of e Then each comp onent of

H f B i s cr itical But s ince H i s bipartite each of the s e comp onents i s also bipartite

It follows that all comp onents of H f B are tr ivial If p oss ible let h b e an e dge of H

which also has b oth its ends in B Then by s imple counting we can s ee that any p erfect

matching through h contains f but not e Thi s contradicts the hyp othe s i s that e and f are

equivalent Now supp os e that H fe f g i s connecte d Then there i s an even path P in

H fe f g connecting the two ends of e But then P e i s an o dd circuit in H which i s

imp oss ible Therefore we conclude that H fe f g i s di sconnecte d

Pro of of the Theorem Let v b e a vertex of degree in G As in the pro of of Theorem

there exi st di sjoint minimal class e s Q Q Q in G each dep ending on an e dge

1 2 

incident with v Each Q i s e ither a s ingleton a removable e dge or a doubleton Thus in

i

order to prove that there are at least s ingletons it suce s to prove that there are at

most two doubletons Assume to the contrary that there are three di stinct doubletons say

Q fe e g Q fe e g and Q fe e g We shall de duce that in thi s cas e G must

1 1 1 2 2 2 3 3 3

C e ither b e K or

6 4

We know that for i G Q i s a bipartite matching covere d graph Let us wr ite

i

H G Q denote the bipartition of H by A B By Lemma H Q and H Q

1 2 3

are di sconnecte d Note that for each i each comp onent of H Q i s a bipartite

i

graph with a p erfect matching and thus its partition has parts of equal cardinality

Cas e The two e dge s of Q b elong to dierent comp onents of H Q In thi s

3 2

cas e the e dge s e and e of Q must b elong to dierent comp onents of H Q Since

2 2 2 3

e must b e cut e dge s of the comp onents of H Q Thus H Q i s di sconnecte d e and

2 3 2 2

G Q Q Q has four comp onents Let us denote them by H H H and H Now

1 2 3 1 2 3 4

s ince G i s connecte d we must have that the graph obtaine d f rom G by shr inking each H

j

to a s ingle vertex i s a K with Q Q and Q as its p erfect matchings For any H and

4 1 2 3 j

any M notation as in all vertice s but one of H are matche d among thems elve s So

i j

Ear decomp os itions of matching covere d graphs

jV H j i s o dd It i s easy to s ee that the larger part of the bipartition of H i s a barr ier of

j j

G It follows that each jV H j and hence that G K

j 4

Cas e The two e dge s of Q b elong to the same comp onent of H Q Let J and

3 2

J b e the comp onents of H Q and let A V J A B V J B A V J A

2 1 1

1

e of and B V J B We may assume without loss of generality that the e dge s e and

3 3

1

e of Q must b elong to the same Q have the ir ends in J In thi s cas e the e dge s e and

2 2 3 2

comp onent of H Q

3

Let L and L b e the comp onents of H Q and assume without loss of generality that

3

J i s a subgraph of L Then Q i s a e dge cut of L One comp onent of L Q i s J let

2 2

A B denote the bipartition of the other comp onent of L Q where A A V L A

2 2 2 2 1

and B B V L B Let A B denote the bipartition of L where A A V L

2 1 3 3 3

and B B V L Thus G Q Q Q has three comp onents with bipartitions

3 1 2 3

A B A B and A B Furthermore jA j jB j jA j jB j and jA j jB j

1 1 2 2 3 3 1 1 2 2 3 3

As in the pro of of Theorem for each i there i s a p erfect matching M of G which

i

i i Thus for i and contains b oth the e dge s of Q but i s di sjoint f rom Q

i i

j jV Q A j jV Q B j where V Q i s the s et of end vertice s of the e dge s

i j i j i

in Q Now Since G i s connecte d an e dge of Q must join vertice s in A and A and the

i 1 1 3

other e dge of Q must join vertice s in B and B Thus in summary we have

1 1 3

An e dge of Q say e joins a vertex u in A with a vertex v in A and the other

1 1 1 1 1 3

e dge e of Q joins a vertex u in B with a vertex v in B

1 1 1 1 1 3

an e dge of Q say e joins a vertex u in A with a vertex v in B and the other

2 2 2 1 2 2

e of Q joins a vertex u in B with a vertex v in A and e dge

2 2 2 1 2 2

an e dge of Q say e joins a vertex u in A with a vertex v in B and the other

3 3 3 2 3 3

e dge e of Q joins a vertex u in B with a vertex v in A

3 3 3 2 3 3

Figure shows all the incidence s de scr ib e d ab ove

e

1

v v u v u u

1 2 3 3 1 2

A e A e A

1 3 3 2 2

e

3

e

2

B B B

1 3 2

v u u u v v

3 1 2 3 1 2

e

1

Figure

C Now we shall show that all A and B are s ingletons and thereby de duce that G i s

6 i i

Towards thi s end we shall rst show that u u Supp os e that u u Since G i s

1 2 1 2

Carvalho Luccche s i and Murty

bicr itical G fu u g has a p erfect matching N Thi s p erfect matching nece ssar ily contains

1 2

e and e Thus u u

1 2 1 2

Let N denote the s et of e dge s in N which have at least one end in A B n fu u g

1 1 1 2

v g and let Let M denote the subs et of e dge s of M matching vertice s in A B n fv

1 1 3 3 1

1

M denote the subs et of e dge s of M matching vertice s in A B n fv v g Then M

2 2 2 2 2

2

N fe e g M M i s a p erfect matching of G which contains b oth Q and Q Thi s

1 2 1 2

1 2

i s imp oss ible b ecaus e Q i s a minimal class induce d by one e dge incident with the vertex v

1

of maximum degree and Q i s a minimal class induce d by a dierent e dge incident with the

2

vertex v

u u It now follows that A i s a barr ier of G But s ince Thus u u and s imilarly

1 2 1 1 2

G i s a br ick jA j Similar arguments show that in f act all A and B are s ingletons It

1 i i

follows that G i s C

6

Remark The ab ove theorem i s b e st p oss ible For example the br ick in Figure has exactly

one removable e dge

Canonical e ar decomp o s itions

In thi s s ection we shall pre s ent a pro of of Theorem We start with the following us eful

lemma

Lemma Let G be a matching covered graph Let C rS be a nontrivial tight cut

of G Suppose that GfS sg has a canonical ear decomposition with d double ears and that

Gf S sg is bipartite Then G also has a canonical ear decomposition with d double ears

Pro of If p oss ible cho os e a counterexample G S such that i jV Gj jE Gj i s as small

as p oss ible and ii sub ject to i jS j i s as small as p oss ible

It follows f rom i that G has no multiple e dge s Since C i s a nontr ivial jS j i s at least

three Let us rst cons ider the cas e in which jS j

S sg has exactly four vertice s s b e ing one of them Let Cas e jSj In thi s cas e Gf

u v and w b e the other three vertice s where u i s of degree in Gf S sg with v and w as its

ne ighb ours

Let P where

P G G G Q G G Q

1 2 1 1 r r 1 r 1

b e a canonical ear decomp os ition of GfS sg with d double ears For convenience we shall

assume that the e dge of G i s not incident with s We shall s ee how an ear decomp os ition

1

P of G can b e obtaine d f rom P by mo difying some of the ears in it

If P i s a path in GfS sg which do e s not contain s as an inter nal vertex then there i s

a path P in G with E P E P Let us refer to P as the path that corre sp onds to P

All mo dications of ears except one cons i st of merely replacing paths in the ears in P by

the paths in G that corre sp ond to them In the exceptional cas e a path in an ear of P i s

mo die d so as to include the e dge s v u and uw which are the two e dge s of G not in GfS sg

A preci s e de scr iption of thi s mo dication i s given b elow

Ear decomp os itions of matching covere d graphs

Let G b e the rst graph in P which contains the vertex s Then the path P containing

i+1

s in the ear Q must have s as an inter nal vertex Then e ither i the two e dge s in P incident

i

with s are incident with the dierent vertice s in G or ii the s e two e dge s are incident with

the same vertex in G We treat the s e two cas e s s eparately

i Supp os e that the two e dge s in P incident with s are incident with the dierent vertice s

in G In thi s cas e mo dify P by replacing the vertex s by v u w

ii Supp os e the two e dge s in P incident with s are incident with the same vertex in G say

v then do not mo dify the ear Q except for replacing the paths in it by the corre sp onding

i

paths in G Any subs equent ear in P which contains a path R incident with s would have

s as an end vertex of R Cons ider the rst o ccurence of such an ear say Q where the e dge

k

of R incident with s i s an e dge xw incident with w in G Mo dify the path R by replacing

x w in R by x w u v

For i r let Q b e the path or the pair of paths obtaine d by mo difying Q

i

i

according to the ab ove sp ecie d rule s Cons ider the s equence

P G G G G Q G G Q

1

1 2 1 1 r r 1 r 1

of subgraphs of G Then it i s easy to ver ify that P i s a canonical ear decomp os ition of G

with d double ears

S sg i s a brace and let e b e any e dge of Cas e jSj Firstly let us supp os e that Gf

Gf S sg which i s not incident with s and hence not b elonging to the cut C Then by

S sg Since e i s not in S e i s removable in G as well Let us wr ite e i s removable in Gf

G G Then by hyp othe s i s G fS sg GfS sg has a canonical ear decomp os ition and

G fS sg i s bipartite By induction G has a canonical ear decomp os ition It can b e extende d

to a canonical ear decomp os ition of G by adding e at the end as a s ingle ear

If not Gf S sg i s not a brace then it has a nontr ivial tight cut D rT where T i s a

prop er subs et of S But D i s also a tight cut of G The pro of can now b e complete d by two

applications of the induction hyp othe s i s

A matching covere d graph G i s ne arbipartite if there are two e dge s e and f of G such

that i G fe f g i s a bipartite graph with bipartition A B and ii e has b oth its ends

in A and f has b oth its ends in B We shall refer to H as the underlying bipartite

graph and the e dge s e and f as sp ecial e dge s In any ear decomp os ition of a bipartite

matching covere d graph the rst nonbipartite memb er of the s equence i s a nearbipartite

graph Thus by studying ear decomp os itions of nearbipartite matching covere d graphs we

are able to de duce theorems concer ning ear decomp os itions of nonbipartite matching covere d

graphs

Lemma Every nearbipartite matching covered graph has a canonical ear decomposition

which uses exactly one double ear

Pro of By induction on jV j jE j We may assume that jV j

If G has a nontr ivial tight cut C then it i s also a tight cut of H Furthermore one of the

C contractions of G i s bipartite and the other i s a nearbipartite graph The pro of follows

f rom induction and Lemma

Carvalho Luccche s i and Murty

C then there i s nothing more So we may assume that G i s a br ick If G i s e ither K or

6 4

to b e prove d If not by G has a removable e dge say h Clearly h cannot b e e ither e

or f By induction G h has a canonical ear decomp os ition which us e s exactly one double

ear We can add h as a s ingle ear at the end to obtain an ear decomp os ition of G which us e s

exactly one double ear

Pro of of Theorem Let G b e any graph which has an ear decomp os ition

P G G G G

1 2 r

which us e s d double ears Then we wi sh to prove that G has a canonical ear decomp os ition

P which also us e s exactly d double ears In order to prove thi s it suce s to show that

the rst nonbipartite graph in P has a canonical ear decomp os ition which us e s exactly one

double ear Let G b e the rst nonbipartite graph in P Then G i s bipartite and G i s

i i1 i

obtaine d f rom G by adding a double ear P P where P i s an o dd path with ends in

i1 1 2 1

one part of the bipartition of G and P i s an o dd path with ends in the other part of the

i1 2

bipartition of G If we replace P and P by e dge s e and f we obtain a nearbipartite

i1 1 2

matching covere d graph say G To prove the theorem it clearly suce s to show that G has

a canonical ear decomp os ition which us e s exactly one ear But thi s i s valid by

Reference s

M H Carvalho and C L Lucche s i Matching covere d graphs and o dd sub divi

s ions of K and C J Combinatorial Theory B

4 6

M H Carvalho and C L Lucche s i Removable e dge s in br icks Pro cee dings of

the Rio conference

Marcelo H Carvalho Decomp o s icao Otima em Orelhas Para Grafo s

Matching Covere d PhD the s i s submitte d to the Univers ity of Campinas Bras il

Marcelo H Carvalho and Claudio L Lucche s i Optimal ear decomp os itions of

matching covere d graphs Under preparation

L Lovasz Ear decomp os itions of matching covere d graphs Combinatorica

L Lovasz and M D Plummer Matching Theory NorthHolland

L Lovasz Matching structure and the matching lattice J Combinatorial Theory

B

L Lovasz and SVempala Under preparation

USR Murty The Matching Lattice and Relate d Topics Preliminary Re

p ort Univers ity of Waterlo o Waterlo o Canada

Ear decomp os itions of matching covere d graphs

C Little and F Rendl An algor ithm for the ear decomp os ition of a f actor

covere d graph J Austral Math Soc Ser ie s A

Z Szigeti The two ear theorem on matchingcovere d graphs Submitte d for pub

lication