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 graphGconsists in a host treeT and a collection
(
Tv)
v∈V(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
andv
′are adjacent inGif and only if thecorre-sponding subtreesTvandTv′have at leasttvertices in common inT. The class of graphs that have an
(
h,
s,
t)
-representationis 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
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 vertexv
, represented byNG(v)
, is the set of vertices adjacentto
v
. Theclosed neighborhoodNG[
v
]
isNG(v)
∪ {
v
}
. Thedegreeofv
, denoted bydG(v)
, is the cardinality ofNG(v)
. Forsimplicity, 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 vertexv
∈
V(
G)
or an edgee∈
E(
G)
is acritical elementofG ifχ(
G−
v) < χ(
G)
orχ(
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)
v∈V(G)of subpaths of a host treeT satisfying thattwo vertices
v
andv
′ofGare adjacent if and only ifPvandPv′ 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 notionof full representation atq.
Let
⟨
P,
T⟩
be a VPT representation ofGand letqbe a vertex ofT with degreeh. The connected components ofT−
qare 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 branchesTiandTjarelinkedby 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 existpathsPv
,
Pw,
Pu∈
Psuch that: (i) the branchesTiandTjare linked byPv; (ii)Pwis contained inTiand intersectsPvin atleast 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 aclique ofG.
The following theorem from [1] shows that any VPT representation which is not full at some vertexqofTwithdT
(
q)
≥
4can 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 twobranches 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)
, anddT′
(
x)
=
3 if x=
q′;
h
−
1 if x=
q;
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 verticesuandv
are adjacent inB(
G/
C)
if and only if(1) u
v
̸∈
E(
G)
;(2) there exists a vertexx
∈
Csuch thatxu∈
E(
G)
andxv
∈
E(
G)
; (3) there exists a vertexy∈
Csuch thatyu∈
E(
G)
andyv
̸∈
E(
G)
; (4) there exists a vertexz∈
Csuch thatzu̸∈
E(
G)
andzv
∈
E(
G)
.It is clear that ifC
∈
C(
G)
andv
∈
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)
andv
∈
V(
G)
−
C :(
i)
Ifv
̸∈
V(
B(
G/
C))
then B((
G−
v)/
C)
=
B(
G/
C)
;(
ii)
ifv
∈
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 andv
are adjacent in B(
G/
C)
, then Puand 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 cliqueofG. We say that a vertexsisa dominated stable vertexifs
∈
Sand there existss′∈
Ssuch thatN(
s)
⊆
N(
s′)
. Notice that ifGis 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)
, thenxy
̸∈
E(
B(
G/
N[
s]
))
becausexy∈
E(
G)
and, ify∈
S′thenxy̸∈
E(
B(
G/
N[
s]
))
becauseN(
y)
⊆
N[
x]
. Then, the chromaticnumber 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)
. Thusχ(
B(
G/
N[
s]
))
≤
χ(
B(
G/
K))
. Hence maxC∈C(G)(χ(
B(
G/
C)))
=
χ(
B(
G/
K))
, that is,Kis a principalclique 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 thatG̸∈ [
h,
2,
1]
andG−
v
∈ [
h,
2,
1]
for every vertexv
∈
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 letv
̸∈
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. Thusχ(
B(
G/
C))
−
1≤
hand henceχ(
B(
G/
C))
≤
h+
1. Then, byTheorem 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)
ifv
∈
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)∩
K̸=
N(
s2)∩
K .Proof. ByTheorem 11,G
∈ [
h+
1,
2,
1]
. Then, byTheorem 7, sinceKis a principal clique ofG, we have thatχ(
B(
G/
K))
=
h+
1.
(1)(i) It is clear thatV
(
B(
G/
K))
⊆
V(
G)
−
K. Suppose there existsv
∈
V(
G)
−
Ksuch thatv
̸∈
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)
ish-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 ofB
(
G/
K)
andχ(
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 existsv
∈
V(
B(
G/
K))
such thatχ(
B(
G/
K)
−
v)
=
h+
1. Then, sincev
∈
V(
B(
G/
K))
, byClaim 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)∩
Kthens1s2̸∈
E(
B(
G/
K))
andNB(G/K)(
s1)=
NB(G/K)(
s2), which will contradictthe fact thatB
(
G/
K)
is(
h+
1)
-vertex critical. Indeed, ifN(
s1)∩
K=
N(
s2)∩
Kthens1s2̸∈
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)
, k2s3̸∈
E(
G)
;k3s1̸∈
E(
G)
,k3s3∈
E(
G)
. And, sinceN(
s1)∩
K=
N(
s2)∩
K, it follows thatk1s2∈
E(
G)
,k2s2∈
E(
G)
,k3s2̸∈
E(
G)
. In addition,s3s2̸∈
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 tosee 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 removedfromPv. 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 canobtain a new VPT representation
⟨
P′,
T′⟩
ofGwithdT′
(
q)
≤
h. Thus, byLemma 6,B(
G/
Cq)
ish-colorable which contradictsthe 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 thatCq
− {
k}
is not a clique ofG−
k, there must existsv
∈
V(
G)
−
Cqsuch thatv
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. Sincev
∈
V(
G)
−
Cq, there existsisuch thatPvis contained inTi. And, sinceh≥
3, there exists a branchTs, withs̸=
1,
2,
i. LetPube the path ofPlinkingTsandTr, withr
̸=
i. It is clear thatu∈
Cqandv
is not adjacent tou, which contradicts the fact thatv
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 ofC
(
G)
,Pis the family(
Pv)
v∈V(G)withPv= {
C∈
C(
G)
|
v
∈
C}
andPvis a subpath ofTfor allv
∈
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)
andq̸∈
V(
Pwi)
. Notice that everyw
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 thatw1
andw2
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 branchgraph, there existsz
∈
Cqsuch thatzw1
∈
E(
G)
,zw3
∈
E(
G)
and, sinceN(w1)
$N(w2)
,zw2
∈
E(
G)
, which implies thatPzcontains 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]
andχ(
B(
G′/
C q))
≤
h.We claim that there exists anh-coloring ofB
(
G′/
Cq
)
such that if there existsx∈
Cqandw
i, w
j∈ {
w1, w2, . . . , w
k}
withx
w
i∈
E(
G)
,xw
j∈
E(
G)
thenIndeed, if
w
iandw
jhave the same color inB(
G′/
Cq)
thenw
iw
j̸∈
E(
B(
G′/
Cq))
. Then we can assume thatN(w
i)
⊆
N(w
j)
,since, by hypothesis, there existsx
∈
Cqsuch thatxw
i∈
E(
G)
andxw
j∈
E(
G)
; thus, for anys̸=
j, no vertex ofCqis adjacentto
w
iandw
s. This implies thatw
iis an isolated vertex ofB(
G′/
Cq)
. Therefore, we can change the color ofw
ito either of theh
−
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′/
Cq
)
satisfying the condition(2).Now, we give anh-coloring, denotedc, ofB
(
G/
Cq)
as follows: givenw
∈
V(
B(
G/
Cq))
, byLemma 5, there exists 1≤
i≤
ksuch 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 ifuv
∈
E(
B(
G/
Cq))
thenc(
u)
̸=
c(v)
.Sinceu
v
∈
E(
B(
G/
Cq))
, byLemma 5,PuandPvare in different branches ofTatqsayTiandTj. Moreover, there existsx∈
Cqsuch thatxu
∈
E(
G)
andxv
∈
E(
G)
, but this implies thatxw
i∈
E(
G)
andxw
j∈
E(
G)
. Hence since our coloring satisfies thecondition(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, ifG
∈
Split thenGhas no dominated stable vertices.Case
(
2)
: Suppose thatG̸∈
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′)
v∈V(G), given by:Pv′
=
Pv if
v
∈
Cq;
{
qv}
ifv
∈
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)
−
Cqandxy
∈
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)
becausein 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))
andxy̸∈
E(
B(
G/
Cq))
thenxy∈
E(
G)
. Thus, without loss of generality, we can assume thatNG
(
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 sinceG̸∈
Split. Sincexy
∈
E(
G)
, we can assume, without loss of generality, thatqxlies on the path ofTbetweenqandqy, which impliesthatxdominatesyinG
˜
.Then, by Case
(
1)
,G˜
is not a minimal non-[
h,
2,
1]
graph. Thus there existsv
∈
V(
G˜
)
such that(
G˜
−
v)
∈ [
h+
1,
2,
1]
. Ifv
∈
V(
B(
G˜
/
Cq))
, thenχ(
B((
G˜
−
v)/
Cq))
=
h+
1. Moreover, byClaim 4and sinceB(
G˜
/
Cq)
=
B(
G/
Cq)
, we have thatB
((
G˜
−
v)/
Cq)
=
B(
G˜
/
Cq)
−
v
=
B(
G/
Cq)
−
v
=
B((
G−
v)/
Cq)
. Henceχ(
B((
G−
v)/
Cq))
=
h+
1 which contradicts the factthatGis 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
. Henceχ(
B((
G˜
−
v)/(
Cq−
v)))
=
h+
1. Moreover, itis easy to see thatB
((
G˜
−
v)/(
Cq−
v))
=
B((
G−
v)/(
Cq−
v))
; thusχ(
B((
G˜
−
v)/(
Cq−
v)))
=
χ(
B((
G−
v)/(
Cq−
v)))
=
h+
1which 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 11χ(
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′∈
Ksuch 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′)/(
Cq
− {
k′}
))
=
B(
G/
Cq)
. Thusχ(
B((
G−
k′)/(
Cq− {
k′}
)))
=
χ(
B(
G/
Cq))
=
h+
1,2. 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′)/(
Cq
− {
k′}
)))
=
V(
B(
G/
Cq))
andE(
B((
G−
k′)/(
Cq− {
k′}
)))
⊆
E(
B(
G/
Cq))
. Letuv
∈
E(
B(
G/
Cq))
such that u
v
̸∈
E(
B((
G−
k′)/(
Cq
− {
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 isw
̸=
usuch thatw
∈
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=
uv
∈
E(
B(
G/
K))
. By the definition of branch graph, there existsk
∈
Ksuch thatku∈
E(
G)
,kv
∈
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
}
, henceN
(
u)
∩
N(v)
̸= ∅
. Since there are no dominated stable vertices,N(
u)
̸⊆
N(v)
,N(v)
̸⊆
N(
u)
. Thusuv
∈
E(
B(
G/
K))
. Hence for eachk∈
Kthere existu,v
∈
Ssuch thatuv
∈
E(
B(
G/
K))
. Observe that ifk˜
∈
Ksuch thatk˜
̸=
k, thenN(
k˜
)
∩
S̸=
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
=
uv
by any edge ofB(
G/
K)
, we will prove thatχ(
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 soχ(
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)
,k2y̸∈
E(
G)
, k3x̸∈
E(
G)
,k3y∈
E(
G)
. Notice thatk1̸=
kbecause every vertex ofKhas exactly two neighbors inSande̸=
xy. Therefore, ifk2̸=
kandk3̸=
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 ofkinSareuandv
, 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)
andk4̸=
k,k4̸=
k1. Ifk4y∈
E(
G)
, thenN[
k1] =
N[
k4] = {
x,
y} ∪
Kwhich means thatk1andk4are true twins contradicting the minimality ofG. Hencek4y̸∈
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 thatv
iv
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.
Notice that the vertices ofGHare partitioned into a stable setSHof sizen
= |
V(
H)
|
corresponding to the verticesv
i, anda central cliqueKHof size
|
E(
H)
| + |{
v
∈
V(
H)
|
dH(v)
=
1}|
corresponding to the remaining vertices. The usefulness ofGHrelies 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) ofTheorem 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 allv
∈
V(
GH)
. First, ifv
=
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 cliqueKH, byLemma 9andTheorem 7, we have thatGH
−
v
i∈ [
h,
2,
1]
. Secondly, ifv
=
v
ijwheree=
v
iv
j∈
E(
H)
, as a directconsequence of the way in which the graphGHwas obtained fromH, we have thatB
((
GH−
v
ij)/(
KH− {
v
ij}
))
=
H−
e. Thenχ(
B((
GH−
v
ij)/(
KH− {
v
ij}
)))
=
χ(
H−
e)
. And,χ(
H−
e)
=
hbecauseHis(
h+
1)
-critical. HenceGH−
v
ij∈ [
h,
2,
1]
. SinceHis 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)
-criticalgraphsH.
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; thenB
(
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
iv
j∈
E(
H)
. It is clear thatv
ik∈
E(
G)
andv
jk∈
E(
G)
. Moreover, by item (ii) ofTheorem 12, there existk′
,
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′andkare true twins inG, which contradicts the fact thatGis minimal non-
[
h,
2,
1]
graph. Hencek′̸=
k′′. Thusk′v
j̸∈
E(
G)
andk′′
v
i
̸∈
E(
G)
. Therefore,v
iv
j∈
E(
H)
.Hence we can define a functionf that assigns to each vertexk
∈
Kan edgev
iv
j∈
E(
H)
, that is, an element ofKH. Notethat inGHthe vertex
v
ij∈
KHis adjacent exactly tov
iandv
j. Hence the functionfcan be extended to a new functionf˜
fromK
∪
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 allv
∈
V(
G)
. Moreover, in [4] it was proved thatG−
v
∈
EPT for allv
∈
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)
:G̸∈
VPT. SinceGis a minimal non-[
h,
2,
1]
graph, thenG−
v
∈ [
h,
2,
1]
for allv
∈
V(
G)
. ThusG−
v
∈
VPT for allv
∈
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.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.