The recognition of the class of indecomposable digraphs under low hemimorphy

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Discrete Mathematics 309 (2009) 3404–3407

Contents lists available atScienceDirect

Discrete Mathematics

journal homepage:www.elsevier.com/locate/disc

Note

The recognition of the class of indecomposable digraphs under low hemimorphy

A. Boussaïri

^a

, P. Ille

^b,^∗

aFaculté des Sciences Aïn Chock, Département de Mathématiques et Informatique, Km 8 route d’El Jadida, BP 5366 Maarif, Casablanca, Maroc

bInstitut de Mathématiques de Luminy, CNRS – UMR 6206, 163 avenue de Luminy, Case 907, 13288 Marseille Cedex 09, France

a r t i c l e i n f o

Article history:

Received 3 February 2006

Received in revised form 17 April 2008 Accepted 29 August 2008

Available online 24 September 2008 Keywords:

Interval

Indecomposable digraph Hemimorphy

a b s t r a c t

Given a digraphG=(^V,Â), the subdigraph ofGinduced by a subsetXofVis denoted by G[X]. With each digraphG=(^V,Â)is associated its dualG^?=(^V,Â^?)defined as follows:

for anyx,^y∈_V_,(^x,^y)∈_A^?_if(^y,^x)∈A. Two digraphsGandHare hemimorphic ifGis isomorphic toHor toH^?. Givenk >0, the digraphsG = (^V,Â)ând^H = (^V,^B)âre^k- hemimorphic if for everyX⊆V, with|X| ≤k,G[X]andH[X]are hemimorphic. A classC of digraphs isk-recognizable if every digraphk-hemimorphic to a digraph ofCbelongs to C. In another vein, given a digraphG=(^V,Â)^{, a subset}^Xôf^Vis an interval ofGprovided that fora,^b∈Xandx∈V−X,(â,^x)∈Aif and only if(^b,^x)∈A, and similarly for(^x,â) and(^x,^b). For example,∅,{x}, wherex∈V, andVare intervals called trivial. A digraph is indecomposable if all its intervals are trivial. We characterize the indecomposable digraphs which are 3-hemimorphic to a non-indecomposable digraph. It follows that the class of indecomposable digraphs is 4-recognizable.

1. Introduction

Adirected graphor simplydigraph Gconsists of a finite and nonempty setVofverticestogether with a prescribed collection Aof ordered pairs of distinct vertices, called the set of thearcsofG. Such a digraph is denoted by

(

^V

,

^A

)

. For example, given a setV,

(

^V

, ∅ )

^{is the}^empty^{digraph on}^V^whereas

(

^V

, (

^V

×

V

) − { (

^x

,

^x

) ;

x

∈

V

} )

^{is the}^complete^{digraph on}V. Given a digraph G

= (

^V

,

^A

)

, with each nonempty subsetXofVassociate thesubdigraph

(

^X

,

^A

∩ (

^X

×

X

))

^of^G^{induced by}^X^{denoted by}^G

[

X

]

. In another respect, given digraphsG

= (

^V

,

^A

)

^and^G⁰

= (

^V⁰

,

^A⁰

)

, a bijectionf fromVontoV⁰is anisomorphismfromGonto G⁰provided that for anyx

,

^y

∈

V,

(

^x

,

^y

) ∈

Aif and only if

(

^f

(

^x

),

^f

(

^y

)) ∈

A⁰. Two digraphs are thenisomorphicif there exists an isomorphism from one onto the other. Finally, a digraphH embedsinto a digraphGifHis isomorphic to a subdigraph of G.

With each digraphG

= (

^V

,

^A

)

associate itsdual G^?

= (

^V

,

^A^?

)

^{and its}complement G

= (

^V

,

^A

)

defined as follows. Given x

6=

y

∈

V,

(

^x

,

^y

) ∈

A^?if

(

^y

,

^x

) ∈

A, and

(

^x

,

^y

) ∈

Aif

(

^x

,

^y

) 6∈

A. The digraph

− →

G

= (

^V

, − →

A

)

is then defined by

− →

A

=

A

−

A^?. Given digraphsGandH,GandHarehemimorphicifGis isomorphic toHorH^?. Given an integerk

>

0, consider digraphs G

= (

^V

,

^A

)

^and^H

= (

^V

,

^B

)

. The digraphsGandHarek-hemimorphicif for every subsetXofV, with

|

X

| ≤

k, the subdigraphs G

[

X

]

andH

[

X

]

are hemimorphic. A digraphGisk-forced(up to duality) ifGandG^?are the only digraphsk-hemimorphic to G.

We need some notations. LetG

= (

^V

,

^A

)

be a digraph. Forx

6=

y

∈

V,x

−→

_Gyory

←−

_Gxmeans

(

^x

,

^y

) ∈

Aand

(

^y

,

^x

) 6∈

A, x

←→

_Gymeans

(

^x

,

^y

), (

^y

,

^x

) ∈

Aandx

· · ·

_Gymeans

(

^x

,

^y

), (

^y

,

^x

) 6∈

A. Forx

∈

VandY

⊆

V,x

−→

_GYsignifies that for every

∗Corresponding author.

E-mail addresses:aboussairi@hotmail.com(A. Boussaïri),ille@iml.univ-mrs.fr(P. Ille).

doi:10.1016/j.disc.2008.08.023

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A. Boussaïri, P. Ille / Discrete Mathematics 309 (2009) 3404–3407 3405

y

∈

Y,x

−→

_Gy. ForX

,

^Y

⊆

V,X

−→

_GY signifies that for everyx

∈

X,x

−→

_GY. Forx

∈

V and forX

,

^Y

⊆

V,x

←−

_GY, x

←→

_GY,x

· · ·

_GY,X

←→

_GYandX

· · ·

_GYare defined in the same way. Furthermore, an equivalence relation, denoted by

≡

_G, between the ordered pairs of distinct vertices of a digraphG

= (

^V

,

^A

)

is defined in the following way. Forx

6=

y

∈

V and foru

6= v ∈

V,

(

^x

,

^y

) ≡

_G

(

^u

, v)

if the function, which attributesutoxand

v

^toy, is an isomorphism fromG

[{

x

,

^y

}]

ontoG

[{

u

, v }]

. Equivalently,

(

^x

,

^y

) ≡

_G

(

^u

, v)

^if^x

−→

_Gyandu

−→

_G

v

^or^x

←−

_Gyandu

←−

_G

v

^or^x

←→

_Gyandu

←→

_G

v

^or x

· · ·

_Gyandu

· · ·

_G

v

. The negation is denoted by

(

^x

,

^y

) 6≡

_G

(

^u

, v)

. Given a subsetXofV, an elementxofV

−

Xis aseparator ofXif there existu

, v ∈

Xsuch that

(

^x

,

^u

) 6≡

_G

(

^x

, v)

. The set of the separators ofXis denoted byS_G

(

^X

)

^.

A digraphG

= (

^V

,

^A

)

^{is a}^posetprovided that forx

,

^y

,

^z

∈

V, ifx

−→

_Gyandy

−→

_Gz, thenx

−→

_Gz. With each poset Q

= (

^V

,

^A

)

associate itscomparability digraph C

(

^Q

) = (

^V

,

^A

∪

A^?

)

. Given a digraphG

= (

^V

,

^A

)

, distinct verticesxandyof Gform adirectedpair if eitherx

−→

_Gyory

−→

_Gx. A digraph is atournament if all its pairs are directed. For example,

( {

0

,

¹

,

²

} , { (

⁰

,

¹

), (

¹

,

²

), (

²

,

⁰

) } )

is a tournament, called 3-cycleand denoted byC₃. Atotal order is both a poset and a tournament. Given a total orderO

= (

^V

,

^A

)

^,^x

<

^y^means^x

−→

_Oyforx

,

^y

∈

V.

Given a digraphG

= (

^V

,

^A

)

^{, a subset}^X^of^V^{is an}^interval[3,7] (or anautonomousset [5,8,9] or aclan[4] or ahomogeneous set [2,6] or amodule[11]) ofGifS_G

(

^X

) = ∅

. For instance,

∅

,Vand

{

x

}

, wherex

∈

V, are intervals ofGcalledtrivialintervals.

A digraph isindecomposable[3,7,10] (orprime[2] orprimitive[4]) if all its intervals are trivial; otherwise, it isdecomposable.

The indecomposability bears a certain rigidity. The next result illustrates this fact in the case of the posets.

Theorem 1 ([5,9]).Let Q

= (

^V

,

^A

)

be an indecomposable poset. For every poset Q⁰

= (

^V

,

^A⁰

)

^{, if C}

(

^Q⁰

) =

C

(

^Q

)

^{, then Q}⁰

=

Q or Q⁰

=

Q^?.

Given a posetQ, any digraphG, 3-hemimorphic toQ, is a poset such thatC

(

^G

) =

C

(

^Q

)

. Therefore, every indecomposable poset is 3-forced. To obtain an analogue ofTheorem 1for the tournaments, the comparability digraph is replaced by theC₃- structure. Given a tournamentT

= (

^V

,

^A

)

, the family of the subsetsXofV, such thatT

[

X

]

is isomorphic toC₃, is called the C₃-structureofTand denoted byC₃

(

^T

)

^.

Theorem 2 ([1]). Let T

= (

^V

,

^A

)

be an indecomposable tournament. For every tournament T⁰

= (

^V

,

^A⁰

)

^{, if C}3

(

^T⁰

) =

C₃

(

^T

)

^, then T⁰

=

T or T⁰

=

T^?.

In other words, every indecomposable tournament is 3-forced. To generalize the two theorems above, we have to disallow the embedding of the following digraph and its dual. The digraph

( {

0

,

¹

,

²

} , { (

⁰

,

²

), (

²

,

⁰

), (

⁰

,

¹

) } )

is denoted byF. The digraphsFandF^?are calledflags. A digraphGis then said to be without flags whenFandF^?do not embed intoG.

Theorem 3 ([1]).An indecomposable digraph without flags is3-forced.

The flags are generalized in the following way. Given an integern

≥

4, consider a permutation

σ

^of

{

0

, . . . ,

ⁿ

−

2

}

. The digraphF_n

(σ )

is defined on

{

0

, . . . ,

ⁿ

−

1

}

in the following manner:

(1) F_n

(σ ) [{

0

, . . . ,

ⁿ

−

2

}]

is the total order

σ(

⁰

) < · · · < σ (

ⁿ

−

2

)

^;

(2) givenm

∈ {

0

, . . . ,

ⁿ

−

2

}

, eithermis even and

(

^m

,

ⁿ

−

1

), (

ⁿ

−

1

,

^m

)

are arcs ofF_n

(σ)

^or^m^{is odd and}

(

^m

,

ⁿ

−

1

), (

ⁿ

−

1

,

^m

)

are not.

Givenn

≥

4,F_n

(

^Id^{0,...,n−2}

)

is simply denoted byF_n(seeFig. 1). Fork

≥

2, the digraphsF_2kandF_2k(resp.F_2k+1and

(

^F2k+1

)

^?) are calledgeneralized flags. By definition,F₃

(

^Id^{0,1}

) =

F. We may verify that for a permutation

σ

^of

{

0

, . . . ,

ⁿ

−

2

}

, wheren

≥

3,F_n

(σ )

is decomposable if and only if there isi

∈ {

0

, . . . ,

ⁿ

−

3

}

such that

σ (

ⁱ

)

^and

σ (

ⁱ

+

1

)

share the same parity. Therefore, the generalized flags are indecomposable. Furthermore, given an indecomposable digraphG, ifIis an interval of

− →

G, then the digraph obtained fromG, by reversing all the arcs included inI, is 3-hemimorphic toG. Sometimes, intervals are created in this way so that the obtained digraph equals neitherGnorG^?. For instance, givenn

≥

4, consider the generalized flagF_nand an integeri

>

0 such that 2i

≤

n

−

2. Clearly,

{

1

, . . . ,

²ⁱ

}

is an interval of

− →

F_n. FromF_n, we obtain by reversing the arcs contained in

{

1

, . . . ,

²ⁱ

}

the digraphF_n

(σ

i

)

^{, where}

σ

iis the permutation of

{

0

, . . . ,

ⁿ

−

2

}

which interchangesjand 2i

−

j

+

1 for 1

≤

j

≤

2i. The pair

{

0

,

²ⁱ

}

forms an interval ofF_n

(σ

i

)

. Consequently, the generalized flags are not 3-forced sinceF_nandF_n

(σ

i

)

differ regarding the indecomposability. Incidently, the problem of the recognition of the class of indecomposable digraphs also occurs. Precisely, givenk

>

^{0, a class}^Cof digraphs isk-recognizableif every digraphk-hemimorphic to a digraph ofCbelongs toCas well. As showing byF_nandF_n

(σ

i

)

, the class of indecomposable digraphs is not 3-recognizable. We reconsider these counter-examples with the following observation:

{

0

, . . . ,

²ⁱ

}

is an interval of

− →

F_n and for everyx

∈ {

0

, . . . ,

ⁿ

−

1

} − {

0

,

²ⁱ

}

, we have

(

^x

,

⁰

) 6≡

_F_n

(

^x

,

²ⁱ

)

if and only if 0

<

^x

<

2i. Generally, consider an indecomposable digraphG

= (

^V

,

^A

)

. Given vertices

α

^and

β

^of^G^{such that}

α −→

_G

β

^{, the pair}

{ α, β }

isweakly separatedif

{ α, β } ∪

S_G

( { α, β } )

is an interval of

− →

G and if

α −→

_GS_G

( { α, β } ) −→

_G

β

. The main result consists of the following characterization.

Theorem 4.Let G be an indecomposable digraph. There exists a decomposable digraph3-hemimorphic to G if and only if G admits a weakly separated pair.

As an immediate consequence, we obtain:

Theorem 5.The class of indecomposable digraphs is4-recognizable.

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3406 A. Boussaïri, P. Ille / Discrete Mathematics 309 (2009) 3404–3407

Fig. 1. The generalized flagF2k.

2. The Gallai decomposition theorem

We begin with a well-known property of the intervals. Given a digraphG

= (

^V

,

^A

)

^{, if}^X^and^Yare disjoint intervals ofG, then

(

^x

,

^y

) ≡

_G

(

^x⁰

,

^y⁰

)

^{for any}^x

,

^x⁰

∈

Xandy

,

^y⁰

∈

Y. This property leads to considerinterval partitionsofG, that is, partitions ofV, all the elements of which are intervals ofG. The elements of such a partitionPbecome the vertices of thequotient G

/

^P

= (

^P

,

^A

/

^P

)

^of^G^by^Pdefined as follows: givenX

6=

Y

∈

P,

(

^X

,

^Y

) ∈

A

/

^P^if

(

^x

,

^y

) ∈

Aforx

∈

Xandy

∈

Y. To state the Gallai decomposition theorem below, we need the following strengthening of the notion of interval. Given a digraph G

= (

^V

,

^A

)

^{, a subset}^X^of^V^{is a}strong interval[5,9] ofGprovided thatXis an interval ofGand for each intervalYofG, we have: ifX

∩

Y

6= ∅

, thenX

⊆

YorY

⊆

X. The family of the maximal strong intervals under inclusion which are distinct fromVis denoted byP

(

^G

)

^.

Theorem 6 ([5,9]).Given a digraph G

= (

^V

,

^A

)

^{, with}

|

V

| ≥

2, the family P

(

^G

)

constitutes an interval partition of G. Moreover, the corresponding quotient G

/

^P

(

^G

)

is a complete digraph or an empty digraph or a total order or an indecomposable digraph.

The next result follows fromTheorem 3.

Corollary 7 ([1]).Given digraphs G and H without flags, if G and H are3-hemimorphic, then P

(

^G

) =

P

(

^H

)

^. 3. Proof ofTheorems 4and5

Lemma 8. Consider 3-hemimorphic digraphs G

= (

^V

,

^A

)

^{and H}

= (

^V

,

^B

)

. Given an interval I of G such that

|

I

| ≥

2, if

−

→

G

[

I

] /

^P

( − →

G

[

I

] )

is not a total order, then I is an interval of H.

Proof. Givenx

∈

V

−

I, sinceIis an interval ofG, we have:x

←→

_GIorx

· · ·

_GIorx

−→

_GIorx

←−

_GI. In the first two instances, it follows from the 2-hemimorphy thatx

←→

_HIorx

· · ·

_HI. In the last two ones, since

− →

G

[

I

] /

^P

( − →

G

[

I

] )

^{is not a} total order,P

( − →

G

[

I

∪ {

x

}] ) = {

I

, {

x

}}

. As

− →

G and

− →

H are 3-hemimorphic digraphs without flags, it follows fromCorollary 7 thatP

( − →

H

[

I

∪ {

x

}] ) = {

I

, {

x

}}

. Consequently, eitherx

−→

_HIorx

←−

_HI.

Corollary 9. Consider 3-hemimorphic digraphs G and H. If G is indecomposable, then for every interval I of H, H

[

I

]

is a total order.

Proof. Consider an intervalIofH. By the previous lemma,

− →

H

[

I

] /

^P

( − →

H

[

I

] )

is a total order. We denote the elements of P

( − →

H

[

I

] )

^by^X1

, . . . ,

^Xqin such a way that

− →

H

[

I

] /

^P

( − →

H

[

I

] )

is the total orderX₁

< · · · <

^Xq. For a contradiction, suppose that there isi

∈ {

1

, . . . ,

^q

}

such that

|

X_i

| ≥

2. SinceIis an interval ofH,X_iis also. It follows from the preceding lemma that

−

→

H

[

X_i

] /

^P

( − →

H

[

X_i

] )

is a total order as well. By interchangingHandH^?, we can assume thati

<

q. By denoting byYthe largest element of

− →

H

[

X_i

] /

^P

( − →

H

[

X_i

] )

, we obtain thatY

∪

X_i+1would be an interval of

− →

H

[

I

]

, which contradicts the fact thatX_iis a strong interval of

− →

H

[

I

]

. Consequently, for eachi

∈ {

1

, . . . ,

^q

}

,

|

X_i

| =

1, that is,H

[

I

]

is a total order.

Theorem 10. Consider3-hemimorphic digraphs G

= (

^V

,

^A

)

^{and H}

= (

^V

,

^B

)

. If G is indecomposable and if H is decomposable, then there exist

α 6= β ∈

V such that

{ α, β }

is an interval of H which is weakly separated in G.

Proof. Given a non-trivial intervalIofH, by the preceding corollary,H

[

I

]

is a total order. Denote by

α

^and

β

^{the first} two elements of this total order, with

α −→

_G

β

^{. Clearly,}

{ α, β }

is an interval ofH. Consider the smallest interval

− →

J of

− →

G containing

α

^and

β

^{. We use}Theorem 6. Firstly, suppose that

− →

G

[ − →

J

] /

^P

( − →

G

[ − →

J

] )

is empty. Since

{ α, β }

is directed, there is an element ofP

( − →

G

[ − →

J

] )

^containing

α

^and

β

, which contradicts the minimality of

− →

J . Secondly, assume that

− →

G

[ − →

J

] /

^P

( − →

G

[ − →

J

] )

is indecomposable. As

− →

G

[ − →

J

]

and

− →

H

[ − →

J

]

are 3-hemimorphic digraphs without flags, it follows from Corollary 7that P

( − →

G

[ − →

J

] ) =

P

( − →

H

[ − →

J

] )

^{. Since}

{ α, β }

is an interval ofH,

{ α, β }

is an interval of

− →

H

[ − →

J

]

. We obtain the same contradiction

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A. Boussaïri, P. Ille / Discrete Mathematics 309 (2009) 3404–3407 3407

because

− →

H

[ − →

J

] /

^P

( − →

H

[ − →

J

] )

is indecomposable byTheorem 3. Therefore,

− →

G

[ − →

J

] /

^P

( − →

G

[ − →

J

] )

is a total order. We denote the elements ofP

( − →

G

[ − →

J

] )

^by^X1

, . . . ,

^Xqin such a way that the corresponding quotient isX₁

< · · · <

^Xq. By the minimality of

− →

J ,

α ∈

X₁and

β ∈

X_q. As previously noticed,P

( − →

H

[ − →

J

] ) = {

X₁

, . . . ,

^Xq

}

and hence

− →

H

[ − →

J

] /

^P

( − →

H

[ − →

J

] )

is a total order as well.

Since

{ α, β }

is an interval ofH,

{ α, β }

is an interval of

− →

H

[ − →

J

]

. AsX₁andX_qare strong intervals of

− →

H

[ − →

J

]

,

{ α, β } =

X₁

∪

X_q or, equivalently,X₁

= { α }

andX_q

= { β }

. To conclude, we verify that

{ α, β }

is weakly separated inG. It suffices to show that for everyx

∈

V

− { α, β }

,

(

^x

, α) 6≡

_G

(

^x

, β)

if and only ifx

∈ − →

J

− { α, β }

. Clearly, ifx

∈ − →

J

− { α, β }

, then

α −→

_Gx

−→

_G

β

and hence

(

^x

, α) 6≡

_G

(

^x

, β)

. Conversely, consider an elementuofV

− − →

J . If

{

u

, α }

is directed, then

(

^u

, α) ≡

_G

(

^u

, β)

^because

−

→

J is an interval of

− →

G. Otherwise,

(

^u

, α) ≡

_G

(

^u

, β)

^because

{ α, β }

is an interval ofH.

The proof of the main result follows.

Proof of Theorem 4. Consider an indecomposable digraphG

= (

^V

,

^A

)

. If there is a decomposable digraph 3-hemimorphic toG, then, byTheorem 10,Gpossesses a weakly separated pair. Conversely, consider

α 6= β ∈

Vsuch that

{ α, β }

is a weakly separated pair ofG. Since

{ α, β } ∪

S_G

( { α, β } )

is an interval of

− →

G,

{ β } ∪

S_G

( { α, β } )

is also. Consequently, by reversing all the arcs contained in

{ β } ∪

S_G

( { α, β } )

, we obtain a digraphHwhich is 3-hemimorphic toG. The pair

{ α, β }

is then an interval ofHand thusHis decomposable.

The next result follows fromTheorem 10.

Corollary 11. Consider 3-hemimorphic digraphs G

= (

^V

,

^A

)

^{and H}

= (

^V

,

^B

)

such that G is indecomposable and H is decomposable. There exists a subset X of V , with

|

X

| =

4, such that G

[

X

]

is indecomposable and H

[

X

]

is decomposable. More precisely, G

[

X

]

is isomorphic to F₄(resp. F₄) and H

[

X

]

is isomorphic to F₄

(σ )

^{(resp. F}4

(σ)

^{), where}

σ

is the permutation of

{

0

,

¹

,

²

}

which interchanges either0and1or1and2.

Proof. ByTheorem 10, there are

α 6= β ∈

V such that

{ α, β }

is an interval ofHwhich is weakly separated inG. If

{ α, β } ∪

S_G

( { α, β } ) =

V, then

{ β } ∪

S_G

( { α, β } )

would be an interval ofG. Consequently,

{ α, β } ∪

S_G

( { α, β } ) 6=

V and hence

{ α, β } ∪

S_G

( { α, β } )

is an interval of

− →

G and not ofG. Therefore, there exists

∈

S_G

( { α, β } )

^and^u

6∈ { α, β } ∪

S_G

( { α, β } )

^, such that

{ α, β,

^s

}

is an interval of

− →

G

[{ α, β,

^s

,

^u

}]

and not ofG

[{ α, β,

^s

,

^u

}]

. It follows that

{ α,

^u

}

,

{

s

,

^u

}

and

{ β,

^u

}

are not directed. For example, assume that

α ←→

_Gu. Sinceu

6∈

S_G

( { α, β } )

^,

β ←→

_Guand, necessarily,s

· · ·

_Gu. Furthermore, G

[{ α, β,

^s

}]

is the total order

α <

^s

< β

^or

β <

^s

< α

^because^s

∈

S_G

( { α, β } )

. In both cases,G

[{ α, β,

^s

,

^u

}]

is isomorphic toF₄. AsGandHare 3-hemimorphic, we have

α ←→

_Hu,s

· · ·

_Hu,

β ←→

_HuandH

[{ α, β,

^s

}]

is a total order. To end, it is sufficient to recall that

{ α, β }

is an intervalH

[{ α, β,

^s

}]

.

Theorem 5is directly deduced. Finally,Corollary 11leads to the following.

Remark 12. To obtainTheorem 5, it is not necessary to assume that the considered digraphsG

= (

^V

,

^A

)

^and^H

= (

^V

,

^B

)

^to be 4-hemimorphic. It suffices to require thatGandHare 2-hemimorphic and that for every subsetXofV, with

|

X

| =

3 or 4, the subdigraphsG

[

X

]

andH

[

X

]

are both indecomposable or not.

Acknowledgement

The authors were supported by the France-Morocco cooperation CNRS/CNRST 2005.

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