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Contents lists available atScienceDirect

Discrete Mathematics

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

Characterizing paths graphs on bounded degree trees by

minimal forbidden induced subgraphs

L. Alcón

a

, M. Gutierrez

a,b

, M.P. Mazzoleni

a,b,∗

aDepartamento de Matemática, Universidad Nacional de La Plata, CC 172, (1900) La Plata, Argentina bCONICET, Argentina

a r t i c l e i n f o Article history:

Received 25 June 2013

Received in revised form 25 August 2014 Accepted 27 August 2014

Available online 20 September 2014 Keywords: Intersection graphs Representations on trees VPT graphs Critical graphs Forbidden subgraphs

a b s t r a c t

An undirected graphGis called a VPT graph if it is the vertex intersection graph of a family of paths in a tree. The class of graphs which admit a VPT representation in a host tree with maximum degree at mosthis denoted by[h,2,1]. The classes[h,2,1]are closed under taking induced subgraphs, therefore each one can be characterized by a family of minimal forbidden induced subgraphs. In this paper we associate the minimal forbidden induced subgraphs for[h,2,1]which are VPT with (color)h-critical graphs. We describe how to obtain minimal forbidden induced subgraphs from critical graphs, even more, we show that the family of graphs obtained using our procedure is exactly the family of VPT minimal forbidden induced subgraphs for[h,2,1]. The members of this family together with the minimal forbidden induced subgraphs for VPT (Lévêque et al., 2009; Tondato, 2009), are the minimal forbidden induced subgraphs for[h,2,1], withh ≥3. By taking h=3 we obtain a characterization by minimal forbidden induced subgraphs of the class VPT∩EPT=EPT∩Chordal= [3,2,2] = [3,2,1](see Golumbic and Jamison, 1985).

©2014 Elsevier B.V. All rights reserved.

1. Introduction

Theintersection graphof a family is a graph whose vertices are the members of the family, and the adjacency between vertices is defined by a non-empty intersection of the corresponding sets. Classic examples are interval graphs and chordal graphs.

Aninterval graphis the intersection graph of a family of intervals of the real line, or, equivalently, the vertex intersection graph of a family of subpaths of a path. Achordal graphis a graph without chordless cycles of length at least four. Gavril [5] proved that a graph is chordal if and only if it is the vertex intersection graph of a family of subtrees of a tree. Both classes have been widely studied [3].

In order to allow larger families of graphs to be represented by subtrees, several graph classes are defined imposing con-ditions on trees, subtrees and intersection sizes [9,10]. Leth,sandtbe positive integers; an

(

h

,

s

,

t

)

-representation of a graph

Gconsists in a host treeT and a collection

(

Tv

)

vV(G)of subtrees ofT, such that (i) the maximum degree ofTis at mosth,

(ii) every subtreeTvhas maximum degree at mosts, and (iii) two vertices

v

and

v

are adjacent inGif and only if the

corre-sponding subtreesTvandTv′have at leasttvertices in common inT. The class of graphs that have an

(

h

,

s

,

t

)

-representation

is denoted by

[

h

,

s

,

t

]

. When there is no restriction on the maximum degree ofTor on the maximum degree of the subtrees, we useh

= ∞

ands

= ∞

respectively. Therefore,

[∞

,

,

1

]

is the class of chordal graphs and

[

2

,

2

,

1

]

is the class of interval graphs. The classes

[∞

,

2

,

1

]

and

[∞

,

2

,

2

]

are called VPT and EPT respectively in [7]; and UV and UE, respectively in [13].

Corresponding author at: Departamento de Matemática, Universidad Nacional de La Plata, CC 172, (1900) La Plata, Argentina. E-mail addresses:liliana@mate.unlp.edu.ar(L. Alcón),marisa@mate.unlp.edu.ar(M. Gutierrez),pia@mate.unlp.edu.ar(M.P. Mazzoleni).

http://dx.doi.org/10.1016/j.disc.2014.08.020

(2)

In [6,14], it is shown that the problem of recognizing VPT graphs is polynomial time solvable. Recently, in [1], generalizing a result given in [7], we have proved that the problem of deciding whether a given VPT graph belongs to

[

h

,

2

,

1

]

is NP-complete even when restricted to the class VPT

Split without dominated stable vertices. The classes

[

h

,

2

,

1

]

,h

2, are closed under taking induced subgraphs; therefore each one can be characterized by a family of minimal forbidden induced subgraphs. Such a family is known only forh

=

2 [11] and there are some partial results forh

=

3 [4]. In this paper we associate the VPT minimal forbidden induced subgraphs for

[

h

,

2

,

1

]

with (color)h-critical graphs. We describe how to obtain minimal forbidden induced subgraphs from critical graphs, even more, we show that the family of graphs obtained using our procedure is exactly the family of VPT minimal forbidden induced subgraphs for

[

h

,

2

,

1

]

. The members of this family together with the minimal forbidden induced subgraphs for VPT (seeFig. 2) which were determined in [12,15], are the minimal forbidden induced subgraphs for

[

h

,

2

,

1

]

, withh

3. Notice that by takingh

=

3 we obtain a characterization by minimal forbidden induced subgraphs of the class VPT

EPT

=

EPT

Chordal

= [

3

,

2

,

2

] =

[

3

,

2

,

1

]

[7].

The paper is organized as follows: in Section2, we provide basic definitions and basic results. In Section3, we give necessary conditions for VPT minimal non-

[

h

,

2

,

1

]

graphs. In Section4, we show a procedure to construct minimal non-

[

h

,

2

,

1

]

graphs. In Section5, we describe the family of all minimal non-

[

h

,

2

,

1

]

graphs.

2. Preliminaries

Throughout this paper, graphs are connected, finite and simple. Thevertex setand theedge setof a graphGare denoted byV

(

G

)

andE

(

G

)

respectively. Theopen neighborhoodof a vertex

v

, represented byNG

(v)

, is the set of vertices adjacent

to

v

. Theclosed neighborhoodNG

[

v

]

isNG

(v)

∪ {

v

}

. Thedegreeof

v

, denoted bydG

(v)

, is the cardinality ofNG

(v)

. For

simplicity, when no confusion can arise, we omit the subindexGand writeN

(v)

,N

[

v

]

ord

(v)

. Two verticesx

,

y

V

(

G

)

are calledtrue twinsifN

[

x

] =

N

[

y

]

.

Acomplete setis a subset of mutually adjacent vertices. Acliqueis a maximal complete set. The family of cliques ofGis denoted byC

(

G

)

. Astable set, also called an independent set, is a subset of pairwise non-adjacent vertices.

A graphGisk-colorableif its vertices can be colored with at mostkcolors in such a way that no two adjacent vertices share the same color. Thechromatic numberofG, denoted by

χ(

G

)

, is the smallestksuch thatGisk-colorable. A vertex

v

V

(

G

)

or an edgee

E

(

G

)

is acritical elementofG if

χ(

G

v) < χ(

G

)

or

χ(

G

e

) < χ(

G

)

respectively. A graphGwith chromatic numberhish-vertex critical(resp.h-critical) if each of its vertices (resp. edges) is a critical element.

AVPT representationofGis a pair

P

,

T

whereP is a family

(

Pv

)

vV(G)of subpaths of a host treeT satisfying that

two vertices

v

and

v

ofGare adjacent if and only ifP

vandPv′ have at least one vertex in common; in such case we say

thatPvintersectsPv′. When the maximum degree of the host tree is at mosththe VPT representation ofGis called an

(

h

,

2

,

1

)

-representation ofG. The class of graphs which admit an

(

h

,

2

,

1

)

-representation is denoted by[h, 2, 1].

Since a family of vertex paths in a tree satisfies the Helly property [2], ifCis a clique ofGthen there exists a vertexqofT

such thatC

= {

v

V

(

G

)

|

q

V

(

Pv

)

}

. On the other hand, ifqis any vertex of the host treeT, the set

{

v

V

(

G

)

|

q

V

(

Pv

)

}

, denoted byCq, is a complete set ofG, but not necessarily a clique. In order to avoid this drawback, we introduce the notion

of full representation atq.

Let

P

,

T

be a VPT representation ofGand letqbe a vertex ofT with degreeh. The connected components ofT

q

are called thebranches of T at q. A path iscontainedin a branch if all its vertices are vertices of the branch. Notice that if

NT

(

q

)

= {

q1,q2, . . . ,qh

}

, thenThas exactlyhbranches atq. The branch containingqiis denoted byTi. Two branchesTiand

Tjarelinkedby a pathPv

P if both verticesqiandqjbelong toV

(

Pv

)

.

Definition 1. A VPT representation

P

,

T

isfull at a vertex qofTif, for every two branchesTiandTjofT atq, there exist

pathsPv

,

Pw

,

Pu

Psuch that: (i) the branchesTiandTjare linked byPv; (ii)Pwis contained inTiand intersectsPvin at

least one vertex; and (iii)Puis contained inTjand intersectsPvin at least one vertex.

A clear consequence of the previous definition is that if

P

,

T

is full at a vertexqofT, withdT

(

q

)

=

h

3, thenCqis a

clique ofG.

The following theorem from [1] shows that any VPT representation which is not full at some vertexqofTwithdT

(

q

)

4

can be modified to obtain a new VPT representation without increasing the maximum degree of the host tree; while decreasing the degree of the vertexq.

Theorem 2 ([1]).Let

P

,

T

be a VPT representation of G. Assume there exists a vertex q

V

(

T

)

with dT

(

q

)

=

h

4and two

branches of T at q which are linked by no path ofP. Then there exists a VPT representation

P′

,

T

of G with V

(

T

)

=

V

(

T

)

∪{

q

}

, q

̸∈

V

(

T

)

, and

dT

(

x

)

=

3 if x

=

q

;

h

1 if x

=

q

;

(3)

Branch graphs defined below are used in the following results to describe intrinsic properties of VPT representations.

Definition 3([7]).LetC

C

(

G

)

. Thebranch graphofGfor the cliqueC, denoted byB

(

G

/

C

)

, is defined as follows: its vertices are the vertices ofV

(

G

)

\

Cwhich are adjacent to at least one vertex ofC. Two verticesuand

v

are adjacent inB

(

G

/

C

)

if and only if

(1) u

v

̸∈

E

(

G

)

;

(2) there exists a vertexx

Csuch thatxu

E

(

G

)

andx

v

E

(

G

)

; (3) there exists a vertexy

Csuch thatyu

E

(

G

)

andy

v

̸∈

E

(

G

)

; (4) there exists a vertexz

Csuch thatzu

̸∈

E

(

G

)

andz

v

E

(

G

)

.

It is clear that ifC

C

(

G

)

and

v

V

(

G

)

C, thenC

C

(

G

v)

. The following claim, whose proof is trivial, describes the branch graph ofG

v

for the cliqueCin terms ofB

(

G

/

C

)

.

Claim 4. Let C

C

(

G

)

and

v

V

(

G

)

C :

(

i

)

If

v

̸∈

V

(

B

(

G

/

C

))

then B

((

G

v)/

C

)

=

B

(

G

/

C

)

;

(

ii

)

if

v

V

(

B

(

G

/

C

))

then B

((

G

v)/

C

)

=

B

(

G

/

C

)

v

.

Lemma 5 ([1]).Let C be a clique of a VPT graph G,

P

,

T

be a VPT representation of G and q be a vertex of T such that C

=

Cq.

If

v

is a vertex of B

(

G

/

C

)

, then Pvis contained in some branch of T at q. If two vertices u and

v

are adjacent in B

(

G

/

C

)

, then Pu

and Pvare not contained in the same branch of T at q.

In [1] we proved the following two results which show that there is a relationship between the VPT graphs that can be represented in a tree with maximum degree at mosthand the chromatic number of their branch graphs.

Lemma 6 ([1]).Let C be a clique of a VPT graph G,

P

,

T

be a VPT representation of G and q be a vertex of T such that C

=

Cq.

If dT

(

q

)

=

h, then B

(

G

/

C

)

is h-colorable.

Theorem 7 ([1]).Let G

VPT and h

4. The graph G belongs to

[

h

,

2

,

1

] − [

h

1

,

2

,

1

]

if and only ifmaxC∈C(G)

(χ(

B

(

G

/

C

)))

=

h. The reciprocal implication is also true for h

=

3.

Definition 8. A cliqueKof a graphGis calledprincipalif max

C∈C(G)

(χ(

B

(

G

/

C

)))

=

χ(

B

(

G

/

K

)).

A graphGissplitifV

(

G

)

can be partitioned into a stable setSand a cliqueK. The pair

(

S

,

K

)

is thesplit partitionofGand this partition is unique up to isomorphisms. The vertices inSare calledstable vertices, andKis called thecentral cliqueof

G. We say that a vertexsisa dominated stable vertexifs

Sand there existss

Ssuch thatN

(

s

)

N

(

s

)

. Notice that if

Gis split thenC

(

G

)

= {

K

} ∪ {

N

[

s

] |

s

S

}

. We will writeSplitfor the class of split graphs.

Lemma 9. If

(

S

,

K

)

is the split partition of G

VPT

Split, then K is a principal clique of G.

Proof. Lets

S, we know thatN

[

s

] ∈

C

(

G

)

. Observe thatV

(

B

(

G

/

N

[

s

]

))

=

(

K

N

(

s

))

S′, withS

= {

x

S

|

N

(

x

)

N

(

s

)

̸=

∅}

. We claim that the vertices ofK

N

(

s

)

are isolated inB

(

G

/

N

[

s

]

)

. Indeed, letx

K

N

(

s

)

; ify

K

N

(

s

)

, then

xy

̸∈

E

(

B

(

G

/

N

[

s

]

))

becausexy

E

(

G

)

and, ify

Sthenxy

̸∈

E

(

B

(

G

/

N

[

s

]

))

becauseN

(

y

)

N

[

x

]

. Then, the chromatic

number ofB

(

G

/

N

[

s

]

)

is equal to the chromatic number of the subgraph ofB

(

G

/

N

[

s

]

)

induced byS′, which clearly is a subgraph ofB

(

G

/

K

)

. Thus

χ(

B

(

G

/

N

[

s

]

))

χ(

B

(

G

/

K

))

. Hence maxC∈C(G)

(χ(

B

(

G

/

C

)))

=

χ(

B

(

G

/

K

))

, that is,Kis a principal

clique ofG.

3. Necessary conditions for VPT minimal non-[h, 2, 1] graphs

In this section we give necessary conditions for being a VPT minimal non-

[

h

,

2

,

1

]

graph; recall that:

Definition 10. Aminimal non-[h,2,1] graphis a minimal forbidden induced subgraph for the class

[

h

,

2

,

1

]

, this means any graphGsuch thatG

̸∈ [

h

,

2

,

1

]

andG

v

∈ [

h

,

2

,

1

]

for every vertex

v

V

(

G

)

.

Theorem 11.Let G

VPT and let h

3. If G is a minimal non-

[

h

,

2

,

1

]

graph, then G

∈ [

h

+

1

,

2

,

1

]

.

Proof. LetC

C

(

G

)

and let

v

̸∈

C. We know thatG

v

∈ [

h

,

2

,

1

]

then, byTheorem 7,

χ(

B

((

G

v)/

C

))

h. ByClaim 4,

χ(

B

((

G

v)/

C

))

χ(

B

(

G

/

C

)

v)

χ(

B

(

G

/

C

))

1. Thus

χ(

B

(

G

/

C

))

1

hand hence

χ(

B

(

G

/

C

))

h

+

1. Then, by

Theorem 7,G

∈ [

h

+

1

,

2

,

1

]

.

Theorem 12.Let K be a principal clique of a VPT minimal non-

[

h

,

2

,

1

]

graph G, with h

3. Then:

(

i

)

V

(

B

(

G

/

K

))

=

V

(

G

)

K ;

(

ii

)

if

v

V

(

G

)

K then

|

N

(v)

K

|

>

1;

(

iii

)

B

(

G

/

K

)

is

(

h

+

1

)

-vertex critical;

(

iv

)

if s1,s2

V

(

G

)

K then N

(

s1)

K

̸=

N

(

s2)

K .

(4)

Proof. ByTheorem 11,G

∈ [

h

+

1

,

2

,

1

]

. Then, byTheorem 7, sinceKis a principal clique ofG, we have that

χ(

B

(

G

/

K

))

=

h

+

1

.

(1)

(i) It is clear thatV

(

B

(

G

/

K

))

V

(

G

)

K. Suppose there exists

v

V

(

G

)

Ksuch that

v

̸∈

V

(

B

(

G

/

K

))

. Thus, byClaim 4,

B

((

G

v)/

K

)

=

B

(

G

/

K

)

. SinceGis a minimal non-

[

h

,

2

,

1

]

graph,G

v

∈ [

h

,

2

,

1

]

and, byTheorem 7,B

((

G

v)/

K

)

is

h-colorable. ThusB

(

G

/

K

)

ish-colorable which contradicts the condition(1).

(ii) By item (i), we know that

v

V

(

B

(

G

/

K

))

; then

|

N

(v)

K

| ≥

1. If

|

N

(v)

K

| =

1,

v

would be an isolated vertex of

B

(

G

/

K

)

and

χ(

B

(

G

/

K

))

=

χ(

B

(

G

/

K

)

v)

. But, byClaim 4andTheorem 7,

χ(

B

(

G

/

K

))

=

χ(

B

(

G

/

K

)

v)

=

χ(

B

((

G

v)/

K

))

=

h, which also contradicts the condition(1).

(iii) By the condition (1),

χ(

B

(

G

/

K

))

=

h

+

1; assume, in order to obtain a contradiction, that B

(

G

/

K

)

is not

(

h

+

1

)

-vertex critical; then there exists

v

V

(

B

(

G

/

K

))

such that

χ(

B

(

G

/

K

)

v)

=

h

+

1. Then, since

v

V

(

B

(

G

/

K

))

, by

Claim 4,

χ(

B

((

G

v)/

K

))

=

χ(

B

(

G

/

K

)

v)

=

h

+

1, which contradicts the fact thatGis a minimal non-

[

h

,

2

,

1

]

graph. (iv) We will see that ifN

(

s1)

K

=

N

(

s2)

Kthens1s2

̸∈

E

(

B

(

G

/

K

))

andNB(G/K)

(

s1)

=

NB(G/K)

(

s2), which will contradict

the fact thatB

(

G

/

K

)

is

(

h

+

1

)

-vertex critical. Indeed, ifN

(

s1)

K

=

N

(

s2)

Kthens1s2

̸∈

E

(

B

(

G

/

K

))

by the definition of branch graph. Moreover, ifs3

NB(G/K)

(

s1)then there existk1,k2,k3

Ksuch that:k1s1

E

(

G

)

,k1s3

E

(

G

)

;k2s1

E

(

G

)

, k2s3

̸∈

E

(

G

)

;k3s1

̸∈

E

(

G

)

,k3s3

E

(

G

)

. And, sinceN

(

s1)

K

=

N

(

s2)

K, it follows thatk1s2

E

(

G

)

,k2s2

E

(

G

)

,k3s2

̸∈

E

(

G

)

. In addition,s3s2

̸∈

E

(

G

)

because in other case there would be an induced 4-cycle

{

s2,k2,k3,s3

}

inG, contradicting the fact thatG

VPT (seeFig. 2). Hences3

NB(G/K)

(

s2); we have proved thatNB(G/K)

(

s1)

NB(G/K)

(

s2). By symmetry, it is easy to

see thatNB(G/K)

(

s2)

NB(G/K)

(

s1).

The following lemma and definition are used in the proof ofTheorem 15which states that any VPT minimal non-

[

h

,

2

,

1

]

graph is split and has no dominated stable vertices.

Lemma 13. Let K be a principal clique of a VPT minimal non-

[

h

,

2

,

1

]

graph G, with h

3. Then, K

− {

k

} ∈

C

(

G

k

)

for all k

K .

Proof. Let

P

,

T

be an

(

h

+

1

,

2

,

1

)

-representation ofGand letq

V

(

T

)

such thatK

=

Cq. We claim that

P

,

T

is full atq.

Indeed, suppose, for a contradiction, that

P

,

T

is not full atq. We can assume, without loss of generality, that ifxis an end vertex of a pathPv

Pthen there exists a pathPu

PintersectingPvonly inx, in other case the vertexxcan be removed

fromPv. This implies that any path ofP linking two branches intersects paths contained in those branches. Hence since

P

,

T

is not full atq, there exist branchesTiandTjofTatqwhich are linked by no path ofP. Then, byTheorem 2, we can

obtain a new VPT representation

P′

,

T

ofGwithd

T

(

q

)

h. Thus, byLemma 6,B

(

G

/

Cq

)

ish-colorable which contradicts

the fact thatCqis a principal clique ofG.

Hence since

P

,

T

is full atq, every pair of branches ofTatqare linked by a path ofP. If there existsk

Cqsuch that

Cq

− {

k

}

is not a clique ofG

k, there must exists

v

V

(

G

)

Cqsuch that

v

is adjacent to all the vertices ofCq

− {

k

}

.

LetT1,T2, . . . ,Th+1be the branches ofTatq. Assume, without loss of generality, thatPkis contained in

{

q

} ∪

T1

T2. Since

v

V

(

G

)

Cq, there existsisuch thatPvis contained inTi. And, sinceh

3, there exists a branchTs, withs

̸=

1

,

2

,

i. LetPu

be the path ofPlinkingTsandTr, withr

̸=

i. It is clear thatu

Cqand

v

is not adjacent tou, which contradicts the fact that

v

is adjacent to all the vertices ofCq

− {

k

}

. ThusCq

− {

k

} ∈

C

(

G

k

)

.

Definition 14. Acanonical VPT representationofGis a pair

P

,

T

whereT is a tree whose vertices are the members of

C

(

G

)

,Pis the family

(

Pv

)

vV(G)withPv

= {

C

C

(

G

)

|

v

C

}

andPvis a subpath ofTfor all

v

V

(

G

)

.

In [13] it was proved that every VPT graph admits a canonical VPT representation.

Theorem 15. Let G be a VPT graph and let h

3. If G is a minimal non-

[

h

,

2

,

1

]

graph, then G

Split without dominated stable vertices.

Proof. Case

(

1

)

: Suppose thatG

Split with split partition

(

S

,

K

)

, andGhas dominated stable vertices. Let

P

,

T

be a canonical VPT representation ofG, and letq

V

(

T

)

such thatK

=

Cq. Assume thatNT

(

q

)

= {

q1,q2, . . . ,qk

}

, withk

>

h,

and callT1,T2, . . . ,Tkto the branches ofTatqcontaining the verticesq1,q2, . . . ,qkrespectively. It is clear that for eachqi,

with 1

i

k, there existsPwi

Psuch thatqi

V

(

Pwi

)

andq

̸∈

V

(

Pwi

)

. Notice that every

w

i

S.

Suppose that S

= {

w1, w2, . . . , w

k

}

. Since G has dominated stable vertices, by item (iv) of Theorem 12 we can assume, without loss of generality, thatN

(w1)

$ N

(w2)

. This means that

w1

and

w2

are not adjacent inB

(

G

/

Cq

)

; thus,

by item (iii) of Theorem 12,NB(G/Cq)

(w1)

̸⊆

NB(G/Cq)

(w2)

. Hence there exists l

V

(

B

(

G

/

Cq

))

− {

w1, w2

}

, such that l

NB(G/Cq)

(w1)

NB(G/Cq)

(w2)

. SinceV

(

B

(

G

/

Cq

))

=

S we can assume thatl

=

w3

. Then, by the definition of branch

graph, there existsz

Cqsuch thatz

w1

E

(

G

)

,z

w3

E

(

G

)

and, sinceN

(w1)

$N

(w2)

,z

w2

E

(

G

)

, which implies thatPz

contains the verticesq1,q2andq3. ThenPzis not a path. This contradicts the fact that

P

,

T

is a VPT representation ofG.

We conclude thatS

=

S

− {

w1, w2, . . . , w

k

} ̸= ∅

. LetG

=

G

S′. Notice thatCq

C

(

G

)

andV

(

B

(

G

/

Cq

))

=

{

w1, w2, . . . , w

k

}

. SinceGis a minimal non-

[

h

,

2

,

1

]

graph, thenG

∈ [

h

,

2

,

1

]

and

χ(

B

(

G

/

C q

))

h.

We claim that there exists anh-coloring ofB

(

G

/

C

q

)

such that if there existsx

Cqand

w

i

, w

j

∈ {

w1, w2, . . . , w

k

}

with

x

w

i

E

(

G

)

,x

w

j

E

(

G

)

then

(5)

Indeed, if

w

iand

w

jhave the same color inB

(

G

/

Cq

)

then

w

i

w

j

̸∈

E

(

B

(

G

/

Cq

))

. Then we can assume thatN

(w

i

)

N

(w

j

)

,

since, by hypothesis, there existsx

Cqsuch thatx

w

i

E

(

G

)

andx

w

j

E

(

G

)

; thus, for anys

̸=

j, no vertex ofCqis adjacent

to

w

iand

w

s. This implies that

w

iis an isolated vertex ofB

(

G

/

Cq

)

. Therefore, we can change the color of

w

ito either of the

h

1 remaining colors. This process can be repeated until we have the desiredh-coloring ofB

(

G

/

Cq

)

.

Hence we consider anh-coloring, sayc, ofB

(

G

/

C

q

)

satisfying the condition(2).

Now, we give anh-coloring, denotedc, ofB

(

G

/

Cq

)

as follows: given

w

V

(

B

(

G

/

Cq

))

, byLemma 5, there exists 1

i

k

such thatPwis contained inTi, we definec

(w)

=

c

(w

i

)

. Notice that, in particular,c

(w

i

)

=

c

(w

i

)

.

We will see thatcis a proper coloring ofB

(

G

/

Cq

)

. That is, we have to see that ifu

v

E

(

B

(

G

/

Cq

))

thenc

(

u

)

̸=

c

(v)

.

Sinceu

v

E

(

B

(

G

/

Cq

))

, byLemma 5,PuandPvare in different branches ofTatqsayTiandTj. Moreover, there existsx

Cq

such thatxu

E

(

G

)

andx

v

E

(

G

)

, but this implies thatx

w

i

E

(

G

)

andx

w

j

E

(

G

)

. Hence since our coloring satisfies the

condition(2),c

(w

i

)

̸=

c

(w

j

)

. Thusc

(

u

)

̸=

c

(v)

. Therefore, our coloring is proper.

Thus we have anh-coloring ofB

(

G

/

Cq

)

which contradicts the fact thatCqis a principal clique ofG. We conclude that, if

G

Split thenGhas no dominated stable vertices.

Case

(

2

)

: Suppose thatG

̸∈

Split. SinceGis a minimal non-

[

h

,

2

,

1

]

graph, byTheorem 11,G

∈ [

h

+

1

,

2

,

1

]

. Let

P

,

T

be an

(

h

+

1

,

2

,

1

)

-representation ofGand letq

V

(

T

)

such thatCqis a principal clique ofG. We obtain fromGa new graphG

˜

withV

(

G

˜

)

=

V

(

G

)

, as follows.

˜

Ghas the

(

h

+

1

,

2

,

1

)

-representation

P′

,

T

,P′

=

(

Pv

)

vV(G), given by:

Pv

=

Pv if

v

Cq

;

{

qv

}

if

v

V

(

G

)

Cq

,

whereqvis the vertex ofPvclosest toq

.

We will prove thatG

˜

is a split graph,G

˜

̸∈ [

h

,

2

,

1

]

andG

˜

has dominated stable vertices.

It is clear thatCqis a clique ofG

˜

. We claim that

(

V

(

G

)

Cq

,

Cq

)

is a split partition ofG

˜

. Indeed, ifx

,

y

V

(

G

)

Cqand

xy

E

(

G

˜

)

thenqx

=

qy. ThusNG

(

x

)

Cq

=

NG

(

y

)

Cqwhich contradicts item (iv) ofTheorem 12.

ByLemma 9,Cqis a principal clique ofG

˜

. Then, to see thatG

˜

̸∈ [

h

,

2

,

1

]

it is enough to see thatB

(

G

˜

/

Cq

)

=

B

(

G

/

Cq

)

because

in such case

χ(

B

(

G

˜

/

Cq

))

=

χ(

B

(

G

/

Cq

))

=

h

+

1, and it follows byTheorem 7.

It is straightforward to show thatV

(

B

(

G

˜

/

Cq

))

=

V

(

B

(

G

/

Cq

))

and thatE

(

B

(

G

/

Cq

))

E

(

B

(

G

˜

/

Cq

))

. On the other hand,

notice that ifxy

E

(

B

(

G

˜

/

Cq

))

andxy

̸∈

E

(

B

(

G

/

Cq

))

thenxy

E

(

G

)

. Thus, without loss of generality, we can assume that

NG

(

x

)

Cq

NG

(

y

)

Cqwhich contradicts the fact thatxandyare adjacent inB

(

G

˜

/

Cq

)

. We conclude thatB

(

G

˜

/

Cq

)

=

B

(

G

/

Cq

)

.

Now, to see thatG

˜

has dominated stable vertices, letxandybe vertices ofV

(

G

˜

)

Cqadjacent inG, they exist sinceG

̸∈

Split. Sincexy

E

(

G

)

, we can assume, without loss of generality, thatqxlies on the path ofTbetweenqandqy, which implies

thatxdominatesyinG

˜

.

Then, by Case

(

1

)

,G

˜

is not a minimal non-

[

h

,

2

,

1

]

graph. Thus there exists

v

V

(

G

˜

)

such that

(

G

˜

v)

∈ [

h

+

1

,

2

,

1

]

. If

v

V

(

B

(

G

˜

/

Cq

))

, then

χ(

B

((

G

˜

v)/

Cq

))

=

h

+

1. Moreover, byClaim 4and sinceB

(

G

˜

/

Cq

)

=

B

(

G

/

Cq

)

, we have that

B

((

G

˜

v)/

Cq

)

=

B

(

G

˜

/

Cq

)

v

=

B

(

G

/

Cq

)

v

=

B

((

G

v)/

Cq

)

. Hence

χ(

B

((

G

v)/

Cq

))

=

h

+

1 which contradicts the fact

thatGis a minimal non-

[

h

,

2

,

1

]

graph.

If

v

Cq, then, byLemma 13,Cq

v

C

(

G

v)

; thereforeCq

v

C

(

G

˜

v)

. ThusG

˜

v

Split with split partition

(

V

(

G

)

Cq

,

Cq

v)

. Then, byLemma 9,Cq

v

is a principal clique ofG

˜

v

. Hence

χ(

B

((

G

˜

v)/(

Cq

v)))

=

h

+

1. Moreover, it

is easy to see thatB

((

G

˜

v)/(

Cq

v))

=

B

((

G

v)/(

Cq

v))

; thus

χ(

B

((

G

˜

v)/(

Cq

v)))

=

χ(

B

((

G

v)/(

Cq

v)))

=

h

+

1

which contradicts the fact thatGis a minimal non-

[

h

,

2

,

1

]

graph. We conclude thatG

Split.

InTheorem 12we gave some necessary conditions on the branch graph with respect to a principal clique of a minimal non-

[

h

,

2

,

1

]

graph. In what follows, inTheorem 16, using the fact that all minimal non-

[

h

,

2

,

1

]

graphs are split without dominated stable vertices and the fact that the central clique of a split graph is principal, we will give more necessary conditions for minimal non-

[

h

,

2

,

1

]

graphs.

Theorem 16.Let G be a VPT graph and let h

3. If G is a minimal non-

[

h

,

2

,

1

]

graph with split partition

(

S

,

K

)

then:

(

i

)

for all k

K ,

|

N

(

k

)

S

| =

2;

(

ii

)

|

E

(

B

(

G

/

K

))

| = |

K

|

;

(

iii

)

B

(

G

/

K

)

is

(

h

+

1

)

-critical.

Proof. ByTheorem 15,G

Split without dominated stable vertices. Let

(

S

,

K

)

be the split partition ofG. ByLemma 9Kis a principal clique ofG, byTheorem 11

χ(

B

(

G

/

K

))

=

h

+

1 and by item (i) ofTheorem 12V

(

B

(

G

/

K

))

=

S.

(i) SinceG

VPT

Split without dominated stable vertices,

|

N

(

k

)

S

| ≤

2 for allk

K. Suppose there existsk

K

such that

|

N

(

k

)

S

|

<

2.

ByTheorem 11,G

∈ [

h

+

1

,

2

,

1

]

. Let

P

,

T

be an

(

h

+

1

,

2

,

1

)

-representation ofGand letq

V

(

T

)

such thatK

=

Cq.

ByLemma 13,Cq

− {

k

} ∈

C

(

G

k

)

.

1. If

|

N

(

k

)

S

| =

0: ThenB

((

G

k

)/(

C

q

− {

k

}

))

=

B

(

G

/

Cq

)

. Thus

χ(

B

((

G

k

)/(

Cq

− {

k

}

)))

=

χ(

B

(

G

/

Cq

))

=

h

+

1,

(6)

2. If

|

N

(

k

)

S

| =

1: We will see thatB

((

G

k

)/(

Cq

− {

k

}

))

=

B

(

G

/

Cq

)

. It is clear, by item (ii) ofTheorem 12,

that V

(

B

((

G

k

)/(

C

q

− {

k

}

)))

=

V

(

B

(

G

/

Cq

))

andE

(

B

((

G

k

)/(

Cq

− {

k

}

)))

E

(

B

(

G

/

Cq

))

. Letu

v

E

(

B

(

G

/

Cq

))

such that u

v

̸∈

E

(

B

((

G

k

)/(

C

q

− {

k

}

)))

. Since

|

N

(

k

)

S

| =

1 we can assume, without loss of generality, that

(

N

(v)

Cq

)

(

N

(

u

)

Cq

)

= {

k

}

. Therefore, we have thatNB(G/Cq)

(v)

= {

u

}

, because if there is

w

̸=

usuch that

w

NB(G/Cq)

(v)

then there is

˜

k

Cqsuch thatk

˜

w

E

(

G

)

,k

˜

v

E

(

G

)

. And, sok

˜

is notk′since

|

N

(

k

)

S

| =

1; thus

˜

ku

E

(

G

)

, because

(

N

(v)

Cq

)

(

N

(

u

)

Cq

)

= {

k

}

. But then

{

u

, v, w

} ⊆

N

(

k

˜

)

S, which contradicts the fact that

|

N

(

k

)

S

| ≤

2 for allk

K.

HencedB(G/Cq)

(v)

=

1, which contradicts the fact thatHis

(

h

+

1

)

-vertex critical.

(ii) First we will prove that

|

E

(

B

(

G

/

K

))

| ≤ |

K

|

. Lete

=

u

v

E

(

B

(

G

/

K

))

. By the definition of branch graph, there exists

k

Ksuch thatku

E

(

G

)

,k

v

E

(

G

)

. Thus for eache

E

(

B

(

G

/

K

))

there existsk

K. Hence, by item (i),

|

E

(

B

(

G

/

K

))

| ≤ |

K

|

. Now we will see that

|

K

| ≤ |

E

(

B

(

G

/

K

))

|

. Letk

K. By item (i),

|

N

(

k

)

S

| =

2. Suppose thatN

(

k

)

S

= {

u

, v

}

, hence

N

(

u

)

N

(v)

̸= ∅

. Since there are no dominated stable vertices,N

(

u

)

̸⊆

N

(v)

,N

(v)

̸⊆

N

(

u

)

. Thusu

v

E

(

B

(

G

/

K

))

. Hence for eachk

Kthere existu,

v

Ssuch thatu

v

E

(

B

(

G

/

K

))

. Observe that ifk

˜

Ksuch thatk

˜

̸=

k, thenN

(

k

˜

)

S

̸=

N

(

k

)

S. Because ifN

(

k

˜

)

S

=

N

(

k

)

S, thenk

˜

andkare true twins inGwhich contradicts the fact thatGis a minimal non-

[

h

,

2

,

1

]

graph. Therefore,

|

K

| ≤ |

E

(

B

(

G

/

K

))

|

.

(iii) Lete

=

u

v

by any edge ofB

(

G

/

K

)

, we will prove that

χ(

B

(

G

/

K

)

e

) < χ(

B

(

G

/

K

))

. Notice it is enough to show that there existsk

Ksuch thatB

(

G

/

K

)

e

=

B

((

G

k

)/(

K

− {

k

}

))

; in fact, sinceGis a minimal non-

[

h

,

2

,

1

]

graph, we have thatG

k

∈ [

h

,

2

,

1

]

, and so

χ(

B

((

G

k

)/(

K

− {

k

}

)))

h

<

h

+

1

=

χ(

B

(

G

/

K

))

.

Letkbe a vertex ofKadjacent to bothuand

v

. Observe that, byLemma 13,K

− {

k

}

is a clique ofG

k.

Since every vertex ofSis adjacent to at least two vertices ofK, it follows thatV

(

B

((

G

k

)/(

K

− {

k

}

)))

=

V

(

B

(

G

/

K

))

=

V

(

B

(

G

/

K

)

e

)

. On the other hand, it is clear that any two vertices adjacent inB

((

G

k

)/(

K

− {

k

}

))

are adjacent inB

(

G

/

K

)

. Then, it remains to see thatE

(

B

(

G

/

K

)

e

)

E

(

B

((

G

k

)/(

K

− {

k

}

)))

.

Letxybe an edge ofB

(

G

/

K

)

e, and letk1,k2,k3be vertices ofKsuch thatk1x

E

(

G

)

,k1y

E

(

G

)

,k2x

E

(

G

)

,k2y

̸∈

E

(

G

)

, k3x

̸∈

E

(

G

)

,k3y

E

(

G

)

. Notice thatk1

̸=

kbecause every vertex ofKhas exactly two neighbors inSande

̸=

xy. Therefore, ifk2

̸=

kandk3

̸=

k, we have thatxy

E

(

B

((

G

k

)/(

K

− {

k

}

)))

and the proof is completed. We can assume without lost of generality thatk2

=

kand, since the only neighbors ofkinSareuand

v

, also, without lost of generality, we can assume thatx

=

u.

Since, by item (iii) ofTheorem 12,B

(

G

/

K

)

is

(

h

+

1

)

-vertex critical, then any vertex ofB

(

G

/

K

)

has degree at leasth. Thus there existsk4

Ksuch thatk4x

E

(

G

)

andk4

̸=

k,k4

̸=

k1. Ifk4y

E

(

G

)

, thenN

[

k1

] =

N

[

k4

] = {

x

,

y

} ∪

Kwhich means thatk1andk4are true twins contradicting the minimality ofG. Hencek4y

̸∈

E

(

G

)

. The existence of verticesk1,k3,k4implies thatxy

E

(

B

((

G

k

)/(

K

− {

k

}

)))

.

4. Building minimal non-[h,2,1] graphs

The construction presented here is similar to that done in [1], and a generalization of that used in [4]. Given a graphH

withV

(

H

)

= {

v1, . . . , v

n

}

, letGHbe the graph with vertices:

V

(

GH

)

=

v

i for each 1

i

n

;

v

ij for each 1

i

<

j

nsuch that

v

i

v

j

E

(

H

)

;

˜

v

i for each 1

i

nwithdH

(v

i

)

=

1

;

and the cliques ofGHareKHandCvifor 1

i

n, where:

KH

= {

v

ij

|

1

i

<

j

n

} ∪ { ˜

v

i

|

1

i

nanddH

(v

i

)

=

1

}

,

Cvi

= {

v

i

} ∪ {

v

ij

|

v

j

NH

(v

i

)

} ∪ { ˜

v

i

|

dH

(v

i

)

=

1

}

.

InFig. 1we offer an example.

(7)

Notice that the vertices ofGHare partitioned into a stable setSHof sizen

= |

V

(

H

)

|

corresponding to the vertices

v

i, and

a central cliqueKHof size

|

E

(

H

)

| + |{

v

V

(

H

)

|

dH

(v)

=

1

}|

corresponding to the remaining vertices. The usefulness ofGH

relies on the properties described in the following lemma used in the proof ofTheorem 18.

Lemma 17 ([1]).

(

i

)

GHis a VPT

Split graph without dominated stable vertices;

(

ii

)

B

(

GH

/

KH

)

=

H.

Theorem 18.Let h

3. The graph GHis a minimal non-

[

h

,

2

,

1

]

graph if and only if H is

(

h

+

1

)

-critical.

Proof. Assume thatGHis a minimal non-

[

h

,

2

,

1

]

graph. By item (ii) ofLemma 17,B

(

GH

/

KH

)

=

H. Hence, by item (iii) of

Theorem 16,His

(

h

+

1

)

-critical.

Now, letHbe an

(

h

+

1

)

-critical graph withV

(

H

)

= {

v1, v2, . . . , v

n

}

. ByLemmas 17and9, maxC∈C(GH)

(χ(

B

(

GH

/

C

)))

=

χ(

B

(

GH

/

KH

))

=

χ(

H

)

=

h

+

1. Hence, byTheorem 7,GH

∈ [

h

+

1

,

2

,

1

] − [

h

,

2

,

1

]

. Let us see thatGH

v

∈ [

h

,

2

,

1

]

for all

v

V

(

GH

)

. First, if

v

=

v

i

V

(

H

)

, usingClaim 4and item (ii) ofLemma 17, we have thatB

((

GH

v

i

)/

KH

)

=

B

(

GH

/

KH

)

v

i

=

H

v

i. Thus sinceHis

(

h

+

1

)

-vertex critical,

χ(

B

((

GH

v

i

)/

KH

))

=

h. SinceGH

v

iis a split graph with central clique

KH, byLemma 9andTheorem 7, we have thatGH

v

i

∈ [

h

,

2

,

1

]

. Secondly, if

v

=

v

ijwheree

=

v

i

v

j

E

(

H

)

, as a direct

consequence of the way in which the graphGHwas obtained fromH, we have thatB

((

GH

v

ij

)/(

KH

− {

v

ij

}

))

=

H

e. Then

χ(

B

((

GH

v

ij

)/(

KH

− {

v

ij

}

)))

=

χ(

H

e

)

. And,

χ(

H

e

)

=

hbecauseHis

(

h

+

1

)

-critical. HenceGH

v

ij

∈ [

h

,

2

,

1

]

. Since

His a critical graph it has no degree 1 vertices, and soGHhas no more vertices.

5. Characterization of minimal non-[h,2,1] graphs

In this section, we give a characterization of VPT minimal non-

[

h

,

2

,

1

]

graphs, withh

3. The main result of this section isTheorem 19which states that the only VPT minimal non-

[

h

,

2

,

1

]

graphs are the graphsGHconstructed from

(

h

+

1

)

-critical

graphsH.

Moreover, inTheorem 20, we show that the family of graphs constructed from

(

h

+

1

)

-critical graphs together with the family of minimal forbidden induced subgraphs for VPT [12,15], is the family of minimal forbidden induced subgraphs for

[

h

,

2

,

1

]

, withh

3.

Theorem 19.Let h

3and let G be a VPT graph. G is a minimal non-

[

h

,

2

,

1

]

graph if and only if there exists an

(

h

+

1

)

-critical graph H such that G

GH.

Proof. The reciprocal implication follows directly applyingTheorem 18.

LetGbe a minimal non-

[

h

,

2

,

1

]

graph. ByTheorem 15, we know thatG

Split without dominated stable vertices. Let

(

S

,

K

)

be a split partition ofG. ByTheorem 11,G

∈ [

h

+

1

,

2

,

1

]

. LetH

=

B

(

G

/

K

)

. By item (iii) ofTheorem 16,His an

(

h

+

1

)

-critical graph. Let us see thatG

GH. LetGH

=

(

SH

,

KH

)

. By item (ii) ofLemma 17,B

(

GH

/

KH

)

=

H; then

B

(

GH

/

KH

)

=

B

(

G

/

K

)

. So, sinceV

(

B

(

GH

/

KH

))

=

V

(

B

(

G

/

K

))

,SH

=

S. Moreover, sinceE

(

B

(

GH

/

KH

))

=

E

(

B

(

G

/

K

))

, by item (ii)

ofTheorem 16,

|

KH

| = |

K

|

and, by item (i) ofTheorem 16,

|

N

(

k

)

S

| =

2 for allk

K. Suppose thatN

(

k

)

S

= {

v

i

, v

j

}

;

we will see that

v

i

v

j

E

(

H

)

. It is clear that

v

ik

E

(

G

)

and

v

jk

E

(

G

)

. Moreover, by item (ii) ofTheorem 12, there exist

k

,

k′′

Ksuch thatk

v

i

E

(

G

)

,k′′

v

j

E

(

G

)

. Ifk

=

k′′then, since

|

N

(

k

)

S

| =

2 for allk

K, we have thatk′andk

are true twins inG, which contradicts the fact thatGis minimal non-

[

h

,

2

,

1

]

graph. Hencek

̸=

k′′. Thusk

v

j

̸∈

E

(

G

)

and

k′′

v

i

̸∈

E

(

G

)

. Therefore,

v

i

v

j

E

(

H

)

.

Hence we can define a functionf that assigns to each vertexk

Kan edge

v

i

v

j

E

(

H

)

, that is, an element ofKH. Note

that inGHthe vertex

v

ij

KHis adjacent exactly to

v

iand

v

j. Hence the functionfcan be extended to a new functionf

˜

from

K

StoKH

SH, being the identity function fromStoSH. Moreover,f

˜

is an isomorphism betweenGandGH.

Theorem 20.Let h

3. A graph G is a minimal non-

[

h

,

2

,

1

]

if and only if G is one of the members of F0, F1, . . . ,F16or G

GH,

where H is an

(

h

+

1

)

-critical graph.

Proof. ByTheorem 19, ifG

GHwhereHis an

(

h

+

1

)

-critical graph, thenGis a minimal non-

[

h

,

2

,

1

]

graph.

IfGis any of the members ofF0, . . . ,F16, thenG

̸∈

VPT andG

v

VPT for all

v

V

(

G

)

. Moreover, in [4] it was proved thatG

v

EPT for all

v

V

(

G

)

. ThusG

v

VPT

EPT

= [

3

,

2

,

1

]

[8], which implies thatG

v

∈ [

h

,

2

,

1

]

. HenceGis a minimal non-

[

h

,

2

,

1

]

graph.

Leth

3 and letGbe a minimal non-

[

h

,

2

,

1

]

graph.

Case

(

1

)

:G

̸∈

VPT. SinceGis a minimal non-

[

h

,

2

,

1

]

graph, thenG

v

∈ [

h

,

2

,

1

]

for all

v

V

(

G

)

. ThusG

v

VPT for all

v

V

(

G

)

. Then,Gis a minimal forbidden induced subgraph for VPT. HenceGis one of the members ofF0,F1, . . . ,F16.

Case

(

2

)

:G

VPT. Then, byTheorem 19,G

GH, whereHis an

(

h

+

1

)

-critical graph.

(8)

Fig. 2. Minimal forbidden induced subgraphs for VPT graphs (the vertices in the cycle marked by bold edges form a clique). See [12,15] for more details.

References

[1]L. Alcón, M. Gutierrez, M.P. Mazzoleni, Recognizing vertex intersection graphs of paths on bounded degree trees, Discrete Appl. Math. 162 (2014) 70–77.

[2]C. Berge, Graphs and Hypergraphs, North-Holland, Amsterdam, 1973.

[3]A. Brandstädt, V.B. Le, J.P. Spinrad, Graph Classes: A Survey, in: SIAM Monographs on Discrete Mathematics and Applications., 1999.

[4] M.R. Cerioli, H.I. Nobrega, P. Viana, A partial characterization by forbidden subgraphs of edge path graphs, in: Proceedings of the 10th Cologne-Twente Workshop on Graphs and Combinatorial Optimization, 2011, pp. 109–112.

[5]F. Gavril, The intersection graphs of subtrees in a tree are exactly the chordal graphs, J. Combin. Theory 16 (1974) 47–56.

[6]F. Gavril, A recognition algorithm for the intersection graphs of paths in trees, Discrete Math. 23 (1978) 211–227.

[7]M.C. Golumbic, R.E. Jamison, Edge and vertex intersection of paths in a tree, Discrete Math. 38 (1985) 151–159.

[8]M.C. Golumbic, M. Lipshteyn, M. Stern, Representing edge intersection graphs of paths on degree 4 trees, Discrete Math. 308 (2008) 1381–1387.

[9]R.E. Jamison, H.M. Mulder, Tolerance intersection graphs on binary trees with constant tolerance 3, Discrete Math. 215 (2000) 115–131.

[10]R.E. Jamison, H.M. Mulder, Constant tolerance intersection graphs of subtrees of a tree, Discrete Math. 290 (2005) 27–46.

[11]C.G. Lekkerkerker, J.C. Boland, Representation of a finite graph by a set of intervals on the real line, Fund. Math. 51 (1962) 45–64.

[12]B. Lévêque, F. Maffray, M. Preissmann, Characterizing path graphs by forbidden induced subgraphs, J. Graph Theory 62 (2009) 369–384.

[13]C.L. Monma, V.K. Wei, Intersection graphs of paths in a tree, J. Combin. Theory (1986) 140–181.

[14]A.A. Schaffer, A faster algorithm to recognize undirected path graphs, Discrete Appl. Math. 43 (1993) 261–295.

Figure

Fig. 1. A graph H and the graph G H .
Fig. 2. Minimal forbidden induced subgraphs for VPT graphs (the vertices in the cycle marked by bold edges form a clique)

References

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