Indecomposability graph and indecomposability recognition

Contents lists available atScienceDirect

European Journal of Combinatorics

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

Indecomposability graph and indecomposability recognition

^✩

A. Boussaïri

^a

, A. Chaïchaâ

^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:

Available online 9 August 2013

a b s t r a c t

Given a digraphG = (^V,^A)^{, a subsetXofV}is an interval ofGif fora,^b ∈ Xandv ∈ V\X,(^a, v) ∈ Aif and only if(^b, v)∈ A, and similarly for(v,^a)^and(v,^b). For instance,∅,Vand{v},v∈V, are intervals ofGcalled trivial. A digraph is indecomposable if all its intervals are trivial. LetG=(^V,^A)be a digraph. Givenv∈V,v is an indecomposability vertex ofGifG[V\ {v}]is indecomposable.

The indecomposability graphI(^G)^ofGis defined onVas follows.

Givenv ̸= w ∈ _V,{v, w}is an edge ofI(^G)^ifG[_V \ {v, w}]_is indecomposable. The following is proved for an indecomposable digraphG=(^V,^A). For every digraphH=(^V,^B)^{, ifGandHhave} the same indecomposability vertices and ifd_I(G)(v)=d_I(H)(v)^for eachv ∈V, thenHis indecomposable. We also study other types of indecomposability recognition.

©2013 Elsevier Ltd. All rights reserved.

1. Introduction

Adigraph Gconsists of a finite and nonemptyvertexsetV

(

^G

)

^{and anarcsetA}

(

^G

)

, where an arc is an ordered pair of distinct vertices. Such a digraph is denoted by

(

^V

(

^G

),

^A

(

^G

))

or simply by

(

^V

,

^A

)

^{. For} example, given a finite and nonempty setV

, (

^V

, (

^V

×

V

) \ { (v, v) : v ∈

V

} )

^{is thecompletedigraph} onV. Given a digraphG

= (

^V

,

^A

)

, with each nonempty subsetXofV, we associate thesubdigraph G

[

X

] = (

^X

,

^A

∩ (

^X

×

X

))

^{ofGinduced by}X. Given a proper subsetXofV

,

^G

[

V

\

X

]

is also denoted byG

−

X, and byG

− v

^wheneverX

= { v }

. Two interesting types of digraphs are tournaments and

✩ This work was supported by the France-Morocco cooperation CNRS/CNRST 2007.

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

0195-6698/$ – see front matter©2013 Elsevier Ltd. All rights reserved.

http://dx.doi.org/10.1016/j.ejc.2013.07.009

symmetric digraphs. A digraphGis atournamentif

|

A

(

^G

) ∩ { (v, w), (w, v) }| =

1 for any

v ̸= w ∈

V.

On the other hand, a digraphGissymmetricif

|

A

(

^G

) ∩ { (v, w), (w, v) }| =

0 or 2 for any

v ̸= w ∈

V. A graph Gis defined by a finite and nonemptyvertexsetV

(

^G

)

^{and anedgesetE}

(

^G

)

, where an edge is an unordered pair of distinct vertices. Such a graph is denoted by

(

^V

(

^G

),

^E

(

^G

))

or simply by

(

^V

,

^E

)

^{. For} example, given a finite and nonempty setV

, (

^V

, ∅ )

^{is theemptygraph onVwhereas}

(

^V

, 

V 2

 )

^{is the} completegraph. With a partition

{

V₁

,

^V2

}

ofV, whereV₁

̸= ∅

andV₂

̸= ∅

, we associate thecomplete bipartitegraph

(

^V

, {{ v

1

, v

2

} : v

1

∈

V₁

, v

2

∈

V₂

} )

^{. LetG}be a graph. Given

v ∈

V

(

^G

)

^{, the}neighbourhood N_G

(v)

^of

v

^inGis the family of vertices

w

^{ofGsuch that}

{ v, w } ∈

E

(

^G

)

. The cardinality ofN_G

(v)

is called thedegreeof

v

and is denoted byd_G

(v)

. We say that

v

^{is anisolatedvertex ofGwhenever} d_G

(v) =

0. With each nonempty subsetXofV

(

^G

)

, we associate thesubgraph G

[

X

] = (

^X

,

^E

(

^G

) ∩



X 2

 )

ofGinduced byX. A nonempty subsetCofV

(

^G

)

^{is a}connected componentofGif for anyx

∈

C and y

∈

V

(

^G

) \

C

, {

x

,

^y

} ̸∈

E

(

^G

)

and if for anyx

̸=

y

∈

C, there is a sequencex₀

=

x

, . . . ,

^xn

=

yof elements ofCsuch that

{

x_i

,

^xi+1

} ∈

E

(

^G

)

^{for 0}

≤

i

≤

n

−

1.

Given digraphsGand H, a bijection f fromV

(

^G

)

^ontoV

(

^H

)

^{is an}isomorphismfromGontoH provided that for anyx

,

^y

∈

V

(

^G

), (

^x

,

^y

) ∈

A

(

^G

)

if and only if

(

^f

(

^x

),

^f

(

^y

)) ∈

A

(

^H

)

. Two digraphs are thenisomorphicif there exists an isomorphism from one onto the other. LetGbe a digraph. Forx

̸=

y

∈

V

(

^G

)

^{and foru}

̸= v ∈

V

(

^G

)

^{, we write}

[

x

,

^y

] ≡ [

u

, v ]

if the function

{

x

,

^y

} −→ {

u

, v }

, defined byx

→

u andy

→ v

, is an isomorphism fromG

[{

x

,

^y

}]

ontoG

[{

u

, v }]

; otherwise, we write

[

x

,

^y

] ̸≡ [

u

, v ]

.

Given a digraphG, a subsetIofV

(

^G

)

^{is aninterval}[3,8] (or anautonomousset [5,9,12] or aclan[4]

or ahomogeneousset [2,6] or amodule[15]) ofGprovided that for anya

,

^b

∈

Iand

v ∈

V

(

^G

) \

I, we have

[

a

, v ] ≡ [

b

, v ]

. For instance,

∅ ,

^V

(

^G

)

^and

{ v }

, where

v ∈

V

(

^G

)

, are intervals ofGcalledtrivial intervals. A digraph isindecomposable[3,8,14] (orprime[2] orprimitive[4]) if all its intervals are trivial;

otherwise, it isdecomposable. Given a digraphG, anindecomposabilityvertex ofGis an element

v

^of V

(

^G

)

^{such thatG}

− v

is indecomposable. We denote byI

(

^G

)

the set of the indecomposability vertices ofG. An indecomposable digraph iscriticalif it does not admit indecomposability vertices. Critical and indecomposable digraphs were characterized by Schmerl and Trotter [14].

Given two digraphsG

= (

^V

,

^A

)

^andH

= (

^V

,

^B

)

, we consider an integerksuch that 0

<

^k

< |

V

|

. We say thatGandHare

{−

k

}

-hypomorphicifG

−

XandH

−

Xare isomorphic for every subsetXofVwith

|

X

| =

k. A digraphGis

{−

k

}

-reconstructibleif any digraph

{−

k

}

-hypomorphictoGis isomorphic to it.

Ulam [17] conjectured that a symmetric digraph with at least three vertices is

{−

1

}

-reconstructible.

Then, Pouzet [1, Problem 24] asked whether a digraph is

{−

k

}

-reconstructible. It follows from results of Lopez and Rauzy [10,11] and of Pouzet [13] that a digraph is

{−

k

}

-reconstructible fork

≥

4. For k

=

1, Stockmeyer [16] built an infinite family of non

{−

1

}

-reconstructible tournaments. However, all these tournaments are indecomposable. So the conjecture is still open for decomposable digraphs.

Ille [7] showed that two

{−

1

}

-hypomorphic digraphsG

= (

^V

,

^A

)

^andH

= (

^V

,

^B

)

^{, with}

|

V

| ≥

11, are both indecomposable or not. He introduced the indecomposability graph as follows. Given a digraph G, theindecomposabilitygraph ofGis the graphI

(

^G

)

defined on the setV

(

^G

)

in the following way.

Given

v ̸= w ∈

V

, { v, w } ∈

E

(

I

(

^G

))

^ifG

− { v, w }

is indecomposable. Clearly, two

{−

1

}

-hypomorphic digraphs share the same indecomposability vertices. However, for any finite and nonempty setV, there are digraphsG

= (

^V

,

^A

)

^andH

= (

^V

,

^B

)

with the same indecomposability vertices, such thatGis indecomposable andHis decomposable. It suffices to consider a critical and indecomposable digraph defined onV and the complete digraph onV. The above argumentation leads us to the following natural definition of the indecomposability recognition. First, two digraphs are said to beequivalent under indecomposabilityor simplyequivalent if both are indecomposable or not. Second, given two digraphsG

= (

^V

,

^A

)

^andH

= (

^V

,

^B

)

, consider an integerksuch that 0

<

^k

< |

V

|

. We say thatG andHare

{−

k

}

-equivalentifG

−

XandH

−

Xare equivalent for every subsetXofVwith

|

X

| =

k.

A digraphGis then said to be

{−

k

}

-recognizableif any digraph

{−

k

}

-equivalent toGis equivalent to it. Similarly, we define theF-recognition of a digraphGfor a familyF of integersmsuch that

−|

V

(

^G

) | <

^m

<

0. Notice that two digraphsGandHare

{−

1

}

-equivalent if and only ifI

(

^G

) =

_I

(

^H

)

^. They are

{−

2

}

-equivalent if and only ifI

(

^G

) =

_I

(

^H

)

^.

As observed above, critical and indecomposable digraphs are not

{−

1

}

-recognizable. In Section4, we give a counter-example to

{−

2

}

-recognition. In Section5, we study the indecomposability vertices

and the indecomposability graph of a decomposable digraph. We deduce that a digraph with at least 7 vertices is

{−

2

, −

1

}

-recognizable. In the last section, we establish the following theorem.

Theorem 1.Consider digraphs G

= (

^V

,

^A

)

^{and H}

= (

^V

,

^B

)

^{, with}

|

V

| ≥

9. If I

(

^G

) =

_I

(

^H

)

^{and if} d_I(G)

(v) =

d_I(H)

(v)

^{for every}

v ∈

V , then G and H are equivalent.

Lastly, we prove that this theorem implies the following result which is more general than the aforementioned theorem of Ille [7].

Corollary 2. Given an integer k

≥

1, two

{−

k

}

-hypomorphic digraphs with at least k

+

8vertices are equivalent.

Whenk

≥

4, Corollary 2is an immediate consequence of the fact that digraphs are

{−

k

}

- reconstructible (see [10,11,13]).

2. Indecomposable digraphs

We recall some properties of indecomposable digraphs.

Proposition 3 (Cournier, Ille [3]).Let G

= (

^V

,

^A

)

be an indecomposable digraph, with

|

V

| ≥

3. For each

v ∈

V , there exists a subset X of V such that

v ∈

X

, |

X

| ∈ {

3

,

⁴

,

⁵

}

and G

[

X

]

is indecomposable.

To obtain indecomposable subdigraphs of a larger size, we utilize the next proposition. Given a digraphG

= (

^V

,

^A

)

, we consider a proper subsetXofVsuch that

|

X

| ≥

3 andG

[

X

]

is indecomposable.

We use the following subsets ofV

\

X.

•

Ext

(

^X

)

is the set of

v ∈

V

\

Xsuch thatG

[

X

∪ { v }]

is indecomposable.

• ⟨

X

⟩

is the set of

v ∈

V

\

Xsuch thatXis an interval ofG

[

X

∪ { v }]

.

•

For eachu

∈

X

,

^X

(

^u

)

is the set of

v ∈

V

\

Xsuch that

{

u

, v }

is an interval ofG

[

X

∪ { v }]

. The family constituted by Ext

(

^X

), ⟨

X

⟩

andX

(

^u

)

^{, whereu}

∈

X, is denoted byp_X.

Lemma 4 (Ehrenfeucht and Rozenberg [4]).Given a digraph G

= (

^V

,

^A

)

, consider a proper subset X of V such that

|

X

| ≥

3and G

[

X

]

is indecomposable. The family p_Xrealizes a partition of V

\

X . Moreover, the following assertions hold.

1. Given u

∈

X , for

v ∈

X

(

^u

)

^and

w ∈

V

\ (

^X

∪

X

(

^u

))

^{, if G}

[

X

∪ { v, w }]

is decomposable, then

{

u

, v }

is an interval of G

[

X

∪ { v, w }]

.

2. For

v ∈ ⟨

X

⟩

and

w ∈

V

\ (

^X

∪ ⟨

X

⟩ )

^{, if G}

[

X

∪ { v, w }]

is decomposable, then X

∪ { w }

is an interval of G

[

X

∪ { v, w }]

.

3. For

v ̸= w ∈

Ext

(

^X

)

^{, if G}

[

X

∪ { v, w }]

is decomposable, then

{ v, w }

is an interval of G

[

X

∪ { v, w }]

. The next result follows from the preceding lemma.

Proposition 5 (Ehrenfeucht and Rozenberg [4]).Given an indecomposable digraph G

= (

^V

,

^A

)

, if X is a subset of V such that

|

X

| ≥

3

, |

V

\

X

| ≥

2and G

[

X

]

is indecomposable, then there are

v ̸= w ∈

V

\

X such that G

[

X

∪ { v, w }]

is indecomposable. More precisely, we have

1. Given u

∈

X , if X

(

^u

) ̸= ∅

, then there exist

v ∈

X

(

^u

)

^and

w ∈

V

\ (

^X

∪

X

(

^u

))

such that G

[

X

∪ { v, w }]

is indecomposable;

2. If

⟨

X

⟩ ̸= ∅

, then there exist

v ∈ ⟨

X

⟩

and

w ∈

V

\ (

^X

∪ ⟨

X

⟩ )

such that G

[

X

∪ { v, w }]

is indecomposable.

By applying several times the previous proposition, we obtain the following corollary.

Corollary 6(Ehrenfeucht and Rozenberg [4]).Given an indecomposable digraph G

= (

^V

,

^A

)

, if X is a proper subset of V such that

|

X

| ≥

3and G

[

X

]

is indecomposable, then there exists a subset Y of V such that X

⊆

Y

, |

V

\

Y

| =

1or2and G

[

Y

]

is indecomposable.

After characterizing the critical and indecomposable digraphs, Schmerl and Trotter demonstrated the following theorem.

Theorem 7 (Schmerl, Trotter [14]).The indecomposability graph of an indecomposable digraph, with at least7vertices, is not the empty graph.

Lastly, we recall the following property of an indecomposability graph.

Lemma 8 (Ille [7]).Let G be an indecomposable digraph, with

|

V

(

^G

) | ≥

5. For each

v ∈

V

(

^G

) \

_I

(

^G

)

^, d_I(G)

(v) ≤

2. Furthermore, given

v ∈

V

(

^G

) \

_I

(

^G

)

^{, we have}

1. if

|

N_I(G)

(v) | =

1, then V

(

^G

) \ (

^NI(G)

(v) ∪ { v } )

is an interval of G

− v

^; 2. if

|

N_I(G)

(v) | =

2, then N_I(G)

(v)

is an interval of G

− v

^.

3. Indecomposable digraphs minimal for an unordered pair

Given an indecomposable digraphG

= (

^V

,

^A

)

, consider vertices

v

1

, . . . , v

nofG. The digraphGis minimalfor

{ v

1

, . . . , v

n

}

[3] provided that for every proper subsetW ofV, if

{ v

1

, . . . , v

n

} ⊆

W and if

|

W

| ≥

3, thenG

[

W

]

is decomposable. To describe the indecomposable digraphs minimal for an unordered pair, we utilize the following two families.

Definition 1. Letk

≥

5.

(i) Pkis the family of digraphs defined on

{

1

, . . . ,

^k

}

and satisfying (a) ifi

∈ {

1

, . . . ,

^k

−

2

}

and ifj

∈ {

i

+

2

, . . . ,

^k

}

, then

[

i

,

^j

] ≡ [

1

,

^k

]

; (b) ifi

∈ {

1

, . . . ,

^k

−

1

}

, then

[

i

,

ⁱ

+

1

] ̸≡ [

1

,

^k

]

.

(ii) Qkis the family of digraphs defined on

{

1

, . . . ,

^k

}

and satisfying

(a) ifi

∈ {

1

, . . . ,

^k

−

4

}

and ifj

∈ {

i

+

2

, . . . ,

^k

−

2

}

, then

[

i

,

^j

] ≡ [

1

,

^k

−

2

]

; (b) ifi

∈ {

1

, . . . ,

^k

−

3

}

, then

[

i

,

ⁱ

+

1

] ̸≡ [

1

,

^k

−

2

]

;

(c) ifi

∈ {

1

, . . . ,

^k

−

3

}

, then

[

k

−

1

,

ⁱ

] ≡ [

k

−

1

,

^k

]

and

[

k

,

ⁱ

] ≡ [

k

,

^k

−

2

]

; (d)

[

k

−

1

,

¹

] ̸≡ [

k

−

1

,

^k

−

2

] , [

k

,

¹

] ̸≡ [

k

,

^k

−

1

]

and

[

1

,

^k

−

2

] ≡ [

k

,

^k

−

2

]

.

It follows that the familyPkcontains exactly one symmetric digraphP_k(seeFig. 1, where for i

̸=

_j

∈

_V

(

^Pk

),

ⁱ

←→

_jmeans

(

ⁱ

,

^j

), (

^j

,

ⁱ

) ∈

_A

(

^Pk

)

) and one tournamentT_k(seeFig. 2, where for i

̸=

j

∈

V

(

^Tk

),

ⁱ

−→

jmeans

(

ⁱ

,

^j

) ∈

A

(

^Tk

)

^and

(

^j

,

ⁱ

) ̸∈

A

(

^Tk

)

) admitting the arc

(

¹

,

²

)

. Similarly, the familyQkcontains exactly one symmetric digraphQ_k(seeFig. 3) and one tournamentU_k(seeFig. 4) admitting the arc

(

¹

,

²

)

^.

Proposition 9 (Cournier, Ille [3]).For k

≥

5, the elements ofPk

∪

_Qkare indecomposable and minimal for

{

1

,

^k

}

.

Conversely, we have the following theorem.

Theorem 10 (Cournier, Ille [3]).Given an indecomposable digraph G, with

|

V

(

^G

) | ≥

6, consider distinct vertices

v

^and

w

of G. The digraph G is minimal for

{ v, w }

if and only if there is an isomorphism f from G onto a digraph of the familyPk

∪

_Qk, where k

= |

V

(

^G

) |

, such that f

( { v, w } ) = {

1

,

^k

}

.

We use the following characterization of the indecomposability graph of an indecomposable digraph minimal for an unordered pair.

Lemma 11. Given k

≥

7, consider G

∈

_Pk

∪

_Qk. We haveI

(

^G

) = {

1

,

^k

}

. Moreover, the following holds.

1. If G

∈

_{Pk, then E}

(

I

(

^G

)) = {{

₁

,

²

} , {

₁

,

^k

} , {

_k

−

₁

,

^k

}}

_.

2. If k

≥

8and if G

∈

_Qk, then E

(

I

(

^G

)) = {{

1

,

²

} , {

1

,

^k

} , {

2

,

^k

} , {

k

−

1

,

^k

}}

. If G

∈

_Q7, then either E

(

I

(

^G

)) = {{

1

,

²

} , {

2

,

⁷

} , {

6

,

⁷

}}

or E

(

I

(

^G

)) = {{

1

,

²

} , {

1

,

⁷

} , {

2

,

⁷

} , {

6

,

⁷

}}

.

Fig. 1. The symmetric digraphPk.

Fig. 2. The tournamentTk.

Fig. 3. The symmetric digraphQk.

Fig. 4. The tournamentUk.

Proof. ConsiderG

∈

_Pk

∪

_Qk, wherek

≥

7. ByProposition 9,Gis minimal for

{

1

,

^k

}

and hence I

(

^G

) ⊆ {

₁

,

^k

}

. It follows fromDefinition 1thatG

−

1 is isomorphic to an element ofPk−1

∪

_{Qk−1. By} Proposition 9,G

−

1 is indecomposable. Similarly, ifG

∈

_Pk, thenG

−

k

∈

_Pk−1and henceG

−

kis indecomposable. So assume thatG

∈

_Qk. SetX

= {

1

, . . . ,

^k

−

2

}

. We haveG

[{

1

, . . . ,

^k

−

2

}] ∈

_Pk−2. ByProposition 9,G

[{

1

, . . . ,

^k

−

2

}]

is indecomposable. As

[

k

−

1

,

¹

] ̸≡ [

k

−

1

,

^k

−

2

] ,

^k

−

1

̸∈ ⟨

X

⟩

. For a contradiction suppose thatk

−

1

∈

X

(

ⁱ

)

^{, wherei}

∈ {

1

, . . . ,

^k

−

2

}

. Sincek

−

2

≥

5, we have i

≥

3 ori

≤

k

−

5. Ifi

≤

k

−

5, then

[

k

−

1

,

ⁱ

+

1

] ≡ [

i

,

ⁱ

+

1

]

and

[

k

−

1

,

ⁱ

+

2

] ≡ [

i

,

ⁱ

+

2

]

. As

[

k

−

1

,

ⁱ

+

1

] ≡ [

k

−

1

,

ⁱ

+

2

]

, we should have

[

i

,

ⁱ

+

1

] ≡ [

i

,

ⁱ

+

2

]

. We get a similar contradiction if i

≥

3 by consideringi

−

2 andi

−

1. Consequently,k

−

1

∈

Ext

(

^X

)

^{, that is,G}

−

kis indecomposable.

LetG

∈

_Pk

∪

_Qk, wherek

≥

7. It follows fromProposition 9that

{

1

,

^k

} ∩ {

i

,

^j

} ̸= ∅

for every

{

i

,

^j

} ∈

E

(

I

(

^G

))

. First, assume thatG

∈

_Pk. ByDefinition 1,G

− {

k

−

1

,

^k

} ∈

_Pk−2. ByProposition 9, G

−{

k

−

1

,

^k

}

is indecomposable and hence

{

k

−

1

,

^k

} ∈

E

(

I

(

^G

))

. Symmetrically, we get

{

1

,

²

} ∈

E

(

I

(

^G

))

^. It follows from Definition 1 that G

− {

1

,

^k

}

is isomorphic to an element of Pk−2

∪

_Qk−2. Thus

{

1

,

^k

} ∈

E

(

I

(

^G

))

^{. By}Definition 1,

{

3

, . . . ,

^k

}

is an interval ofG

−

2. So fori

∈ {

3

, . . . ,

^k

} , {

3

, . . . ,

^k

} \ {

i

}

is a non-trivial interval ofG

−{

2

,

ⁱ

}

. ThereforeN_I(G)

(

²

) = {

1

}

. Symmetrically, we getN_I(G)

(

^k

−

1

) = {

k

}

. Now consideri

∈ {

3

, . . . ,

^k

−

2

}

. ByDefinition 1,

{

1

, . . . ,

ⁱ

−

1

}

and

{

i

+

1

, . . . ,

^k

}

are intervals ofG

−

i.

Thus, forj

∈ {

1

, . . . ,

^k

} \ {

i

}

, at least one of

{

1

, . . . ,

ⁱ

−

1

} \ {

j

}

or

{

i

+

1

, . . . ,

^k

} \ {

j

}

is a non-trivial interval ofG

− {

_i

,

^j

}

. Therefore,iis an isolated vertex ofI

(

^G

)

. It follows thatN_I(G)

(

¹

) = {

₂

,

^k

}

_and N_I(G)

(

^k

) = {

1

,

^k

−

1

}

.

Second, assume thatG

∈

_Qk. As already observed,

{

k

−

1

,

^k

} ∈

E

(

I

(

^G

))

^{. By}Definition 1,G

− {

1

,

²

}

is isomorphic to an element ofQk−₂and hence

{

1

,

²

} ∈

E

(

I

(

^G

))

. It follows fromDefinition 1that

{

1

,

^k

}

is an interval ofG

−

2. Thus

{

2

,

^k

} ∈

E

(

I

(

^G

))

^{. ByLemma 8,d}I(G)

(

²

) ≤

2 because 2

̸∈

_I

(

^G

)

^{. Thus} N_I(G)

(

²

) = {

1

,

^k

}

. Leti

∈ {

4

, . . . ,

^k

−

3

}

. Since

{

1

, . . . ,

ⁱ

−

1

}

is an interval ofG

−

i

,

ⁱis an isolated vertex ofI

(

^G

)

^{. By}Definition 1,

{

1

,

²

,

^k

}

is an interval ofG

−

3 and hence 3 is an isolated vertex ofI

(

^G

)

^. ByDefinition 1,

{

1

, . . . ,

^k

−

3

} ∪ {

k

}

is an interval ofG

− (

^k

−

2

)

^{. ThusN}I(G)

(

^k

−

2

) ⊆ {

k

−

1

}

. As

{

1

, . . . ,

^k

−

3

}

is an interval ofG

− {

k

−

2

,

^k

−

1

} ,

^k

−

2 is an isolated vertex ofI

(

^G

)

^{. By}Definition 1,

{

1

, . . . ,

^k

−

2

}

is an interval ofG

− (

^k

−

1

)

. ThereforeN_I(G)

(

^k

−

1

) = {

k

}

. Now assume thatk

≥

8.

Denote byHthe unique digraph such that the bijectionf

: {

2

, . . . ,

^k

} −→ {

1

, . . . ,

^k

−

1

}

, defined by f

(

ⁱ

) =

i

−

1 fori

∈ {

2

, . . . ,

^k

}

, is an isomorphism fromG

−

1 ontoH. ByDefinition 1,H

∈

_Qk−1. By what precedes,I

(

^H

) = {

1

,

^k

−

1

}

. Consequently,I

(

^G

−

1

) =

f⁻¹

{

1

,

^k

−

1

} = {

2

,

^k

}

or equivalently N_I(G)

(

¹

) = {

2

,

^k

}

. It follows thatN_I(G)

(

^k

) = {

1

,

²

,

^k

−

1

}

. Lastly, whenk

=

7, to see that both possibilities for membership of

{

1

,

⁷

}

inE

(

I

(

^G

))

^{note that}

{

1

,

⁷

} ∈

E

(

I

(

^U7

))

^becauseU7

− {

1

,

⁷

}

is indecomposable, while

{

1

,

⁷

} ̸∈

E

(

I

(

^Q7

))

^because

{

3

,

⁶

}

is an interval ofQ₇

− {

1

,

⁷

}

.

The following lemma completes the section.

Lemma 12. Let G

= (

^V

,

^A

)

be an indecomposable digraph, with

|

V

| ≥

9. If there exist a

̸=

b

∈

V

\

_I

(

^G

)

^{such that}

{

a

,

^b

} ∈

E

(

I

(

^G

))

, then G admits an indecomposability vertex which does not belong to N_I(G)

(

^a

) ∪

N_I(G)

(

^b

)

^orI

(

^G

)

possesses an edge which is not included in N_I(G)

(

^a

) ∪

N_I(G)

(

^b

)

^.

Proof. SetX

=

V

\ {

a

,

^b

}

. We haveG

[

X

]

is indecomposable. Ifd_I(G)

(

^a

) =

d_I(G)

(

^b

) =

1, then it follows fromLemma 8thatXis an interval ofG

−

aand ofG

−

b. ThusXwould be an interval ofG. So assume thatd_I(G)

(

^b

) ̸=

1. ByLemma 8, there isu

∈

Xsuch thatN_I(G)

(

^b

) = {

a

,

^u

}

and moreovera

∈

X

(

^u

)

^{. We} distinguish the two cases below.

First, assume thatd_I(G)

(

^a

) =

1, that is,N_I(G)

(

^a

) = {

b

}

. ByLemma 8,b

∈ ⟨

X

⟩

. ByProposition 3 applied toG

[

X

]

, there existsY

⊆

Xsuch thatu

∈

Y

, |

Y

| ∈ {

3

,

⁴

,

⁵

}

andG

[

Y

]

is indecomposable.

As

|

V

| ≥

9, we haveY

⊂

X. ByCorollary 6, there isZ

⊂

X such thatY

⊆

Z

, |

X

\

Z

| =

1 or 2 andG

[

Z

]

is indecomposable. We still havea

∈

Z

(

^u

)

^andb

∈ ⟨

Z

⟩

. SinceG

[

X

∪ {

a

,

^b

}] =

Gis indecomposable, it follows fromLemma 4that

[

b

,

^u

] ̸≡ [

b

,

^a

]

. By the same lemma, we getG

[

Z

∪ {

a

,

^b

}]

is indecomposable as well. Clearly,

|

V

\ (

^Z

∪ {

a

,

^b

} ) | =

1 or 2. If

|

V

\ (

^Z

∪ {

a

,

^b

} ) | =

1, then the unique element ofV

\ (

^Z

∪ {

a

,

^b

} )

is an indecomposability vertex ofGwhich does not belong toN_I(G)

(

^a

) ∪

N_I(G)

(

^b

) = {

a

,

^b

,

^u

}

. If

|

V

\ (

^Z

∪ {

a

,

^b

} ) | =

2, thenV

\ (

^Z

∪ {

a

,

^b

} ) ∈

E

(

I

(

^G

))

^and

(

^V

\ (

^Z

∪ {

_a

,

^b

} )) ∩ (

^NI(G)

(

^a

) ∪

_NI(G)

(

^b

)) = ∅

_.

Second, assume thatd_I(G)

(

^a

) ̸=

1. ByLemma 8, there exists

v ∈

Xsuch thatN_I(G)

(

^a

) = {

b

, v }

and moreoverb

∈

X

(v)

^{. Ifu}

= v

^{, then}

{

u

,

^a

,

^b

}

would be an interval of G. Thus,u

̸= v

^and N_I(G)

(

^a

) ∪

_NI(G)

(

^b

) = {

_a

,

^b

,

^u

, v }

_{. ConsiderY}

⊆

_{Xsuch thatu}

, v ∈

_Y

, |

_Y

| ≥

_{3 andG}

[

_Y

]

is indecom- posable and minimal for

{

u

, v }

. IfY

⊂

X, then we conclude as in the first case. Indeed, byCorollary 6, there isZ

⊂

Xsuch thatY

⊆

Z

, |

X

\

Z

| =

1 or 2 andG

[

Z

]

is indecomposable. We still havea

∈

Z

(

^u

)

andb

∈

Z

(v)

. Furthermore, byLemma 4,

[

a

,

^b

] ̸≡ [

u

, v ]

becauseG

[

X

∪ {

a

,

^b

}] =

Gis indecom- posable. Therefore,G

[

Z

∪ {

a

,

^b

}]

is indecomposable too. We conclude as previously by noticing that

|

V

\ (

^Z

∪ {

a

,

^b

} ) | =

1 or 2. Now, assume thatY

=

X, that is,G

[

X

]

is minimal for

{

u

, v }

. ByTheorem 10, there is an isomorphism fromH

∈

_Pk

∪

_QkontoG

[

X

]

such thatf

( {

1

,

^k

} ) = {

u

, v }

, wherek

= |

V

| −

2.

For instance, assume thatf

(

¹

) =

uandf

(

^k

) = v

. To conclude, we prove that

{

f

(

^k

−

1

), v } ∈

E

(

I

(

^G

))

^.

Fig. 5. The symmetric digraphGk.

Fig. 6. The symmetric digraphHk.

SetW

=

f

( {

1

, . . . ,

^k

−

2

} )

. It follows fromDefinition 1thatH

[{

1

, . . . ,

^k

−

2

}] ∈

_Pk−2. ByProposi- tion 9,H

[{

1

, . . . ,

^k

−

2

}]

is indecomposable becausek

−

2

≥

5. Thus,G

[

W

]

is indecomposable as well.

We still havea

∈

W

(

^u

)

^{inG. Using}Definition 1, we obtain thatk

∈ ⟨{

1

, . . . ,

^k

−

2

}⟩

inH. Therefore, we have inG:

v ∈ ⟨

W

⟩

and henceb

∈ ⟨

W

⟩

becauseb

∈

X

(v)

^{. SinceG}

[

X

∪ {

a

,

^b

}] =

Gis indecompos- able, it follows fromLemma 4that

[

b

,

^a

] ̸≡ [

b

,

^u

]

. By the same lemma applied toG

[

W

]

, we get that G

[

W

∪ {

a

,

^b

}] =

G

− {

f

(

^k

−

1

), v }

is indecomposable.

Lemma 12does not hold for digraphs with 8 vertices. For example, consider the tournament C₈ defined on

{

1

, . . . ,

⁸

}

satisfyingC₈

− {

7

,

⁸

} =

U₆

, {

1

,

⁷

}

is an interval ofC₈

−

8,

{

6

,

⁸

}

is an interval ofC₈

−

7 and

(

¹

,

⁷

), (

⁸

,

⁶

), (

⁸

,

⁷

) ∈

A

(

^C8

)

^{. ByLemma 4,C}8is indecomposable. We verify thatI

(

^C8

) = {

1

,

⁶

}

andE

(

I

(

^C8

)) = {{

1

,

⁶

} , {

1

,

⁸

} , {

6

,

⁷

} , {

7

,

⁸

}}

. Then, choosea

=

7 andb

=

8.

4. Counter-example to

{−

2

}

-recognition

Letk

≥

8. Consider the symmetric digraphG_kdefined onV

(

^Gk

) = {

0

,

¹

, . . . ,

^k

}

byA

(

^Gk

) = { (

ⁱ

,

ⁱ

+

1

), (

ⁱ

+

1

,

ⁱ

) :

2

≤

i

≤

k

−

2

} ∪ { (

⁰

,

^k

−

1

), (

^k

−

1

,

⁰

) } ∪ { (

¹

,

^k

−

3

), (

^k

−

3

,

¹

) } ∪ { (

^k

−

3

,

^k

), (

^k

,

^k

−

3

) }

(seeFig. 5). Clearly,G_k

[{

2

, . . . ,

^k

−

1

} ∪ {

0

}]

is isomorphic toP_k−1. Moreover,

{

1

,

^k

}

is an interval ofG_kand henceG_kis decomposable. Now the symmetric digraphH_kis obtained fromQ_kby adding one vertex 0 and all the possible arcs except

(

⁰

,

^k

−

1

)

^and

(

^k

−

1

,

⁰

)

^{(seeFig. 6).}

SetX

= {

1

, . . . ,

^k

}

. We haveH_k

[

X

] =

Q_k. Clearly, 0

̸∈ ⟨

X

⟩

and it is easy to verify that 0

̸∈

X

(

^u

)

foru

∈

X. It follows fromLemma 4that 0

∈

Ext

(

^X

)

^{, that is,H}kis indecomposable. Furthermore, I

(

^Gk

) =

_I

(

^Hk

) = ( {

0

, . . . ,

^k

} , {{

0

,

¹

} , {

1

,

²

} , {

1

,

^k

} , {

0

,

^k

} , {

2

,

^k

}} )

. Consequently, the digraphG_kis not

{−

2

}

-recognizable. Therefore, the following two problems arise.

Problem 1. Characterize the indecomposable digraphsGsuch that there exist

v ̸= w ∈

V

(

^G

)

^satis- fying each edge of the indecomposability graph ofGintersect

{ v, w }

.

Problem 2. Givenk

>

1, are the digraphs with at least 2k

+

1 vertices

{−

2k

}

-recognizable?

Remark.It follows from the characterization of the critical and indecomposable digraphs due to Schmerl and Trotter [14] that for eachn

≥

4, there exist critical and indecomposable digraphs defined onnvertices. Givenk

≥

0, considern

≥

2k

+

4 and any critical and indecomposable digraph G

= (

^V

,

^A

)

^{such that}

|

V

| =

n. For a contradiction, suppose that there isX

⊂

Vsuch that

|

X

| =

2k

+

1 andG

[

V

\

X

]

is indecomposable. By applyingktimesProposition 5fromG

[

V

\

X

]

, we would obtain x

∈

Xsuch thatG

−

xis indecomposable. Hence, for everyX

⊂

Vsuch that

|

X

| =

2k

+

1

,

^G

[

V

\

X

]

is decomposable. Therefore,Gand the complete digraph onVare

{−

2k

−

1

}

-equivalent without being

equivalent. Consequently, the critical and indecomposable digraphs are not

{−

2k

−

1

}

-recognizable fork

≥

0.

5. Indecomposability graph of decomposable digraphs

In order to characterize the indecomposability graph of decomposable digraphs, we use the following graphs. Letqandrbe positive integers.

•

K_qdenotes the complete graph on

{

1

, . . . ,

^q

}

.

•

K_q,r denotes the complete bipartite graph defined on

{

1

, . . . ,

^q

+

r

}

byE

(

^Kq,r

) = {{

i

,

^j

} :

i

∈ {

1

, . . . ,

^q

} ,

^j

∈ {

q

+

1

, . . . ,

^q

+

r

}}

.

•

L_2,ris defined on

{

1

, . . . ,

^r

+

2

}

byE

(

^L2,r

) =

E

(

^K2,r

) ∪ {{

1

,

²

}}

.

Theorem 13. Given a digraph G

= (

^V

,

^A

)

^{, with}

|

V

| ≥

7

,

G is decomposable if and only if either I

(

^G

) = (

^V

, ∅ )

^orI

(

^G

)

has exactly one connected component C such that

|

C

| ≥

2and one of the following holds:

1. I

(

^G

) = ∅

andI

(

^G

) [

C

]

is isomorphic to K₂

,

^K3

,

^K2,2or K_1,2;

2. there is an isomorphism f from K_1,montoI

(

^G

) [

C

]

such thatI

(

^G

) = {

f

(

¹

) }

, where m

∈ {

1

, . . . ,

|

V

| −

1

}

;

3. there is an isomorphism f from K_2,m ontoI

(

^G

) [

C

]

such that I

(

^G

) = {

f

(

¹

),

^f

(

²

) }

, where m

∈ {

1

, . . . , |

V

| −

2

}

;

4. there is an isomorphism f from L_2,m ontoI

(

^G

) [

C

]

such that I

(

^G

) = {

f

(

¹

),

^f

(

²

) }

, where m

∈ {

1

, . . . , |

V

| −

2

}

;

5. there is an isomorphism f from K₂ontoI

(

^G

) [

C

]

such thatI

(

^G

) = {

f

(

¹

),

^f

(

²

) }

.

Proof. Assume thatGis decomposable and that the indecomposability graph ofGis not the empty graph onV. Considera

̸=

b

∈

Vsuch that

{

a

,

^b

} ∈

E

(

I

(

^G

))

^.

First, assume thatI

(

^G

) = ∅

. SetX

=

V

\ {

a

,

^b

}

. Asa

,

^b

̸∈

_I

(

^G

)

^{, we havea}

,

^b

̸∈

Ext

(

^X

)

^{. Using} Lemma 4, we distinguish the following cases.

•

Assume thata

,

^b

∈ ⟨

X

⟩

. As

|

X

| ≥

5, we haveE

(

I

(

^G

)) = {{

a

,

^b

}}

. Thus,

{

a

,

^b

}

is the only connected component ofI

(

^G

)

which is not reduced to a singleton. Obviously,I

(

^G

) [{

a

,

^b

}]

is isomorphic toK₂.

•

Assume that there isu

∈

Xsuch thata

,

^b

∈

X

(

^u

)

^{. Clearly,}

{

a

,

^b

,

^u

}

is an interval ofGand hence E

(

I

(

^G

)) = {{

a

,

^b

} , {

a

,

^u

} , {

b

,

^u

}}

. Therefore,

{

a

,

^b

,

^u

}

is the only connected component ofI

(

^G

)

which is not reduced to a singleton. Obviously,I

(

^G

) [{

a

,

^b

,

^u

}]

is isomorphic toK₃.

•

Assume that there areu

̸= v ∈

Xsuch thata

∈

X

(

^u

)

^andb

∈

X

(v)

. The only non-trivial intervals ofGare

{

a

,

^u

}

and

{

b

, v }

. It follows thatE

(

I

(

^G

)) = {{

a

,

^b

} , {

a

, v } , {

b

,

^u

} , {

u

, v }}

. Consequently,

{

a

,

^b

,

^u

, v }

is the only connected component ofI

(

^G

)

which is not reduced to a singleton. We obtain thatI

(

^G

) [{

a

,

^b

,

^u

, v }]

is isomorphic toK_2,2.

•

Assume thata

∈ ⟨

X

⟩

andb

∈

X

(v)

^{, where}

v ∈

X. The only non-trivial intervals ofGare

{

b

, v }

and X

∪ {

b

}

. Clearly,E

(

I

(

^G

)) = {{

a

,

^b

} , {

a

, v }}

and hence

{

a

,

^b

, v }

is the only connected component of I

(

^G

)

which is not reduced to a singleton. We getI

(

^G

) [{

a

,

^b

, v }]

is isomorphic toK_1,2.

Second, assume that there exits

α ∈

_I

(

^G

)

^{. SetX}

=

V

\ { α }

. AsGis decomposable,

α ̸∈

Ext

(

^X

)

^. UsingLemma 4, we distinguish the following cases.

•

Assume that

α ∈ ⟨

X

⟩

. We haveI

(

^G

) = { α }

. We also obtain thatE

(

I

(

^G

)) = {{ α, v } : v ∈

_I

(

^G

[

X

] ) }

. Thus,

{ α } ∪

_I

(

^G

[

X

] )

is the only connected component ofI

(

^G

)

which is not reduced to a singleton.

Setm

= |

_I

(

^G

[

X

] ) |

and consider a bijectionf

: {

1

, . . . ,

^m

+

1

} −→ { α } ∪

_I

(

^G

[

X

] )

^{such that} f

(

¹

) = α

^{. Clearly,f}is an isomorphism fromK_1,montoI

(

^G

) [{ α } ∪

_I

(

^G

[

X

] ) ]

.

•

Assume that there isu

∈

X such that

α ∈

X

(

^u

)

. We obtain thatI

(

^G

) = { α,

^u

}

and that E

(

I

(

^G

)) \ {{ α,

^u

}} = {{ α, v } , {

u

, v } : v ∈

_I

(

^G

[

X

] ) \ {

u

}}

. To begin, assume thatI

(

^G

[

X

] ) \ {

u

} ̸= ∅

. Then,

{ α,

^u

} ∪ (

^I

(

^G

[

X

] ) \ {

u

} ) = { α,

^u

} ∪

_I

(

^G

[

X

] )

is the only connected component ofI

(

^G

)

^{which is} not reduced to a singleton. Setm

= |

_I

(

^G

[

X

] ) \ {

u

}|

and consider a bijectionf

: {

1

, . . . ,

^m

+

2

} −→

{ α,

^u

}∪

_I

(

^G

[

X

] )

^{such thatf}

(

¹

) = α

^andf

(

²

) =

u. If

{ α,

^u

} ̸∈

E

(

I

(

^G

))

^{, thenf}is an isomorphism from K_2,montoI

(

^G

) [{ α,

^u

} ∪

_I

(

^G

[

X

] ) ]

. On the other hand, if

{ α,

^u

} ∈

E

(

I

(

^G

))

^{, thenf} is an isomorphism