An interval containment model of a graph maps vertices into intervals of a line in such a way that two vertices are adjacent if and only if the corresponding intervals are comparable under the inclusion relation. Graphs admitting an interval containment model are called containment interval graphs or CI graphs for short. A vertex v of a CI graph G is an end-vertex if there is an interval containment model of G in which the left endpoint of the interval corresponding to v is less than all other endpoints. In this work,we present a characterization of end-vertices in terms of forbidden induced subgraphs.Facultad de Ciencias ExactasConsejo Nacional de Investigaciones Científicas y Técnica

Alcón, Liliana Graciela

Gudiño, Noemí Amalia

Gutiérrez, Marisa

English

SEDICI - Repositorio de la UNLP

Matema´tica Contemporaˆnea, Vol. 45, 86–96
c©2017, Sociedade Brasileira de Matema´tica
End vertices in containment interval
graphs
Liliana Alcon Noemi Gudin˜o Marisa Gutierrez
Abstract
An interval containment model of a graph maps vertices into
intervals of a line in such a way that two vertices are adjacent if
and only if the corresponding intervals are comparable under the
inclusion relation. Graphs admitting an interval containment model
are called containment interval graphs or CI graphs for short. A
vertex v of a CI graph G is an end-vertex if there is an interval
containment model of G in which the left endpoint of the interval
corresponding to v is less than all other endpoints. In this work,
we present a characterization of end-vertices in terms of forbidden
induced subgraphs.
1 Introduction and previous results
A graph G is a containment graph of intervals (or CI for short) if there
is a collection of intervals of the real line (Iw)w∈V (G) satisfying that uw
is an edge of G if and only if Iu ⊂ Iw or Iw ⊂ Iu; the collection is called
a CI model of G. Without loss of generality, it can be assumed that the
intervals of a CI model are closed, with positive length and no two have
2000 AMS Subject Classification: 05C62, 05C75.
Key Words and Phrases: comparability graphs, permutation graphs, containment
models
End vertices in containment interval graphs 87
1 2 2n
1 2 2n 1 2 2n+ 1
1
2
3
2n+ 1
2n
n > 1 n > 2
n > 2 n > 1
J2n+1
Figure 1: These graphs, together with the complement of the graphs in
Figure 2, constitute a minimal family of forbidden induced subgraphs for
comparability graphs.
a same endpoint [5, 8]. Let lw and rw denote, respectively, the left and
the right endpoint of the interval Iw. A vertex v is an end-vertex of G if
there exists a CI model of G where lv < lw for every w ∈ V (G)− {v}.
In this paper, we describe end-vertices of CI graphs using a self - com-
plementary family of forbidden induced subgraphs. In addition, in Section
3, homogeneously representable CI graphs (all vertices are end-vertices)
are characterized by a simple finite family of forbidden induced subgraphs.
A transitive orientation
−→
E of a graph G is an assignment of one of the
two possible directions, −→uw or −→wu, to each edge uw ∈ E(G), such that
if −→uv ∈ −→E and −→vw ∈ −→E then −→uw ∈ −→E . Graphs admitting a transitive
orientation are called comparability graphs [5, 8].
Next theorem, due to Gallai, provides a characterization of compara-
bility graphs in terms of forbidden induced subgraphs. The complement
of a graph G is the graph G =
(
V (G), E(G)
)
such that uv ∈ E(G) if and
only if uv /∈ E(G).
Theorem 1 ([3]). A graph G is a comparability graph if and only if G
contains none of the graphs depicted in Figure 1, nor the complement of
those in Figure 2, as induced subgraphs.
A vertex v of a comparability graph G is a sink (respectively, a source)
88 L. Alcon, N. Gudin˜o and M. Gutierrez
1 2 n 1
2
3
n− 1
n
n > 3
n > 5
1 2 n
n > 2
1 2 n
n > 2
T S
G1
M
G2
G3
Nn
Figure 2: The complement of these graphs, together with the graphs in
Figure 1, constitute a minimal family of forbidden induced subgraphs for
comparability graphs.
End vertices in containment interval graphs 89
if there exists a transitive orientation
−→
E of G such that −→vw /∈ −→E (−→wv /∈ −→E )
for all w ∈ V (G). Olariu and Gimbel, almost simultaneously, obtained
a characterization of sinks in terms of forbidden induced subgraphs as
follows.
Theorem 2 ([4, 6]). Let G be a comparability graph. A vertex v of G is
a sink if and only if G contains none of the graphs A, B, C, D2k+1 with
k ≥ 2, En with n ≥ 3, in Figure 3, with v as the designated vertex.
CI graphs have been widely studied and characterized in different ways.
For instance, they are co-comparability graphs, and in [1, 8] is showed the
relationship between CI graphs and the partially ordered sets (or posets)
of dimension at most 2. Given a poset P = (V,≤), the comparability
graph of P is GP = (V,E) with E = {uv : u < v or v < u in P}. The
dimension of a poset P is the minimal number of linear extension of P
whose intersection is P [1]. The CI graphs are exactly the comparability
graphs of posets with dimension at most 2.
Theorem 3 ([1]). A graph G is CI if and only if G and its complement
G are comparability graphs.
In addition, the class of CI graphs is equivalent to the class of permu-
tation graphs [2, 7].
2 Characterization of end-vertices of CI graphs
In the proof of the main Theorem 5, we will use the following two results.
Remark 1 ([1]). If
−→
E is a transitive orientation of a CI graph G then
there exists a CI model (Iw)w∈V (G) of G compatible with
−→
E ; ie. for each
pair of vertices u,w ∈ V (G), Iu ⊂ Iw if and only if −→uw ∈ −→E .
Given a CI model of a graph G, the right set Rv of a vertex v ∈ V (G)
is {w : lw < rv < rw}; and the left set Lv is {w : lw < lv < rw}. Clearly, if
w ∈ Rv ∩ Lv then Iv ⊂ Iw, and, consequently, w is adjacent to v.
90 L. Alcon, N. Gudin˜o and M. Gutierrez
Lemma 4. Let (Ivi)0≤i≤k be a CI model of a chordless path P = [v0v1 . . . vk]
with k ≥ 2. If Iv1 ⊂ Iv0 then exactly one of the following statements holds:
(i) v2 ∈ Rv0 and rv0 < rvi for 2 ≤ i ≤ k. (ii) v2 ∈ Lv0 and lvi < lv0 for
2 ≤ i ≤ k.
Proof. We proceed by induction. The proposition holds trivially when
k = 2. Let k > 2 and assume Iv1 ⊂ Iv0 . Since v2 is adjacent to v1 and
nonadjacent to v0, we have that v2 ∈ Rv0 \ Lv0 or v2 ∈ Lv0 \Rv0 .
Assume, w.l.g., that v2 ∈ Rv0 . Thus, rv0 < rvi for 2 ≤ i ≤ k − 1. To
complete the proof, we will show that rv0 < rvk .
Since vk is adjacent to vk−1, it follows that Ivk−1 ⊂ Ivk or Ivk ⊂ Ivk−1 .
In the former case, it is clear that rv0 < rvk−1 < rvk . Thus, assume
Ivk ⊂ Ivk−1 ; and, in order to derive a contradiction, suppose that rvk < rv0 .
Since rv0 < rvk−1 and vk−1 is not adjacent to v0, we have that lv0 < lvk−1 .
Therefore lv0 < lvk−1 < lvk , and so Ivk ⊂ Iv0 , which contradicts the fact
that vk and v0 are nonadjacent. 
Theorem 5. Let G be a connected CI graph. A vertex z is an end-vertex
of G if and only if G contains none of the graphs A, A, B, B, C, C,
D2k+1, D2k+1, E3, E2k, E2k, for k ≥ 2, as induced subgraphs with z as
the vertex highlighted in Figure 3.
Proof. Let (Iw)w∈V (G) be a CI model of G with lz < lw for all w ∈
V (G)− {z}. Let −→E be the orientation of G obtained by orienting u to v
(−→uv) whenever Iu ⊂ Iv. Recall that two vertices are adjacent if and only
if the interval corresponding to one of them is contained in the interval
corresponding to the other. Clearly,
−→
E is a transitive orientation of G and
z is a sink; thus, by Theorem 2, G does not contain the graphs A, B, C,
D2k+1 with k ≥ 2 and En with n ≥ 3 as induced subgraph with z as the
designated vertex. Observe that E3 = E3.
Let H be any induced subgraph of G containing vertex z. Denote by
H ′ the graph H plus a vertex z′ adjacent to z, that is, H ′ = (V (H) ∪
z′, E(H) ∪ z′z) and denote by H ′′ the graph H ′ plus a vertex z′′ adjacent
End vertices in containment interval graphs 91
A A¯
B B¯
C C¯
Dn
En E¯n
1 2 3 n
x
1
2
3
n
x
1 2 3 n
x
y
1
2
3
n
n
D¯n
Figure 3: The highlighted vertices in the graphs A, B, C, D2k+1 with
k ≥ 2, En with n ≥ 3, cannot be a sink of a comparability graph. The
highlighted vertices in the graphs A, A, B, B, C, C, D2k+1, D2k+1, E3,
E2k, E2k, for k ≥ 2, cannot be an end-vertex of a CI graph.
92 L. Alcon, N. Gudin˜o and M. Gutierrez
to z′, that is, H ′′ = (V (H ′′) ∪ z′′, E(H) ∪ z′′z′). We claim that H ′ and
H ′′ are CI graphs .
Indeed, CI models ofH ′ andH ′′ can be obtained by adding to (Iw)w∈V (H)
the intervals Iz′ = [lz + , lz + 2] and Iz′′ = [lz − , lz + 3] with  small
enough.
Now, in order to derive a contradiction, assume that H is any one of
the graphs A, B, C, D2k+1, E2k, with k ≥ 2, and z is the vertex of
H highlighted in Figure 3. Notice that (A)′′ = (B)′ = T in Figure 2;
(C)′ = M in Figure 2; (D2k+1)′ = J2k+1 in Figure 1 and E2k ′ = N2k+1 in
Figure 2.
Thus, by the previous claim (2), these graphs are CI, which contradicts
the fact that neither the graphs depicted in Figures 1 and 2 nor their
complements are CI graphs.
To prove the converse, notice first that, by Theorem 2, the vertex z
must be a sink of G, which means there exists a transitive orientation
−→
E
of G in which all edges incident in z are oriented towards z.
Let G′ be the graph G plus a pendent vertex z′ adjacent to z. Since a
transitive orientation
−→
E′ of G′ can be obtained adding
−→
z′z to
−→
E , it follows
that G′ is a comparability graph and z is a sink of G′.
Assume, in order to derive a contradiction, that G′ is not CI. Thus,
G′ contains an induced subgraph H which is either a graph in Figure 2 or
the complement of a graph in Figure 1; even more, H must contain z′, so
H must have a vertex of degree one and z must be the only neighbor of
that vertex.
An inspection of those figures reveals that H has to be one of the graphs
T , S, M , G1, G2, G3 or Nn for some n ≥ 4, in Figure 2; or the complement
of the graph J2k+1 for some k ≥ 2 in Figure 1. In the following paragraph,
we will show that each case implies a contradiction, therefore G′ is a CI
graph.
* If H is any one of the graphs S, G1, G2 or G3, then G contains the
graph A in Figure 3 with z as the highlighted vertex.
* If H is the graph T , then G contains the graph B in Figure 3 with z
End vertices in containment interval graphs 93
as the highlighted vertex.
* If H is the graph M , then we have to consider two cases depending on
which vertex of degree one is z′: its neighbor z is adjacent to two vertices
with degree 3, or its neighbor z is adjacent to a vertex with degree 3 and
other with degree 2. In the first case, G contains the graph A in Figure 3
with z as the highlighted vertex; and, in the latter, G contains the graph
C in Figure 3 with z as the highlighted vertex.
* If H is the graph N4, then G contains the graph E3 in Figure 3 with
z as the highlighted vertex.
* If H is the graph Nn for some n > 4, then, again, we consider two
cases depending on which vertex of degree one is z′: if z′ is the vertex
labelled 1 (or n) then G contains the graphs En−1 and Dn−1 in Figure 3
with z as the highlighted vertex. Notice that En−1 is forbidden when n
is odd and Dn−1 is forbidden when n is even. If z′ is neither the vertex
labelled 1 nor the vertex labelled n, then G contains the graph A in Figure
3 with z as the highlighted vertex.
Thus, we have proved that G′ is CI. By Remark 1, there exist a CI
model (Iw)w∈V (G′) compatible with
−→
E′, i.e. −→uv ∈ −→E′ implies Iu ⊂ Iv.
Clearly, if Iz is an end interval in this model, the vertex z is an end-
vertex of G, and the proof follows.
So, we assume that the left set Lz and the right set Rz of z are non
empty. The fact that z is the only neighbor of z′ implies Lz ∩ Rz = ∅.
Let x be a vertex of Lz minimizing the distance to z in G
′; and Pzx =
[v0 = z, v1, v2, ...vk−1, vk = x] be a shortest path joining z with x. Since z
is a sink, we have that −→v1z ∈
−→
E′, which implies that Iv1 ⊂ Iz. Therefore,
by Lemma 4, v2 ∈ Rz and rz < rvi for every i, or v2 ∈ Lz and lvi < lz for
every i. Since vk = x ∈ Lz, the former is not possible; thus, v2 ∈ Lz which
implies k = 2 and the existence of the induced path [z, v1, x]. Analogously,
there exist vertices v′1 and y ∈ Rz inducing the path [z, v′1, y].
The proof will be complete showing that the concatenation of both
paths induces the graph A with z as the highlighted vertex.
Clearly, x and y are non-adjacent. We claim that x is non-adjacent to
94 L. Alcon, N. Gudin˜o and M. Gutierrez
v′1. Indeed, if x is adjacent to v′1, we have that rv′1 < rx; on the other hand,
since z′ has degree 1 and Iz′ ⊂ Iz, it follows that rx < rz′ and lz′ < ly.
Therefore, lz′ < ly < lv′1 < rv′1 < rx < rz′ which implies Iv′1 ⊂ Iz′ , contrary
to the fact that z′ is adjacent only to z. Analogously, y is non-adjacent to
v1. 
As a consequence of Theorems 2 and 5, we have the following result.
Corollary 6. Let G be a CI graph. A vertex z is an end-vertex of G if
and only if z is a sink of G and G.
3 Characterization of CI homogeneously repre-
sentable graphs
A CI graph G is homogeneously representable if every one of its vertices
is an end-vertex.
Theorem 7. Let G be CI. The graph G is homogeneously representable
if and only if it contains none of the graphs A, A, C, C, E3, depicted in
Figure 3, as induced subgraphs.
Proof. It is a straightforward consequence of Theorem 5 and the facts: A
is an induced subgraph of B and of D2k+1, for k ≥ 2; and E3 is an induced
subgraph of E2k, for k ≥ 2. 
In the previous theorem, the condition that G is a CI graph can be
relaxed by adding C5 to the family of forbidden induced subgraph.
Theorem 8. A graph G is CI homogeneously representable if and only if
it contains none of the graphs C5, A, A, C, C, E3, depicted in Figure 3,
as induced subgraphs.
Proof. The direct implication follows from Theorem 7 and the fact that
C5 is not a comparability graph.
To prove the converse, keep in mind that the graphs depicted in Figures
1 and 2 and their complements are a family of forbidden induced subgraphs
of a CI graph; and notice that C5 and E3 are self-complementary. 
End vertices in containment interval graphs 95
References
[1] B. Dushnik, E. Miller, Partially ordered sets, Amer. J. Math. 63, pp.
600–610, (1941).
[2] S. Even, A. Pnueli, A. Lempel Permutation graphs and transitive
graphs, J. ACM 19, pp. 400–410, (1972).
[3] T. Gallai, Transitiv Orientierbare Graphen, Acta. Math. Hungar. 18,
pp. 25–66, (1967).
[4] J. Gimbel, Source in posets and comparability graphs, Order 9, pp.
361–365,(1992).
[5] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Aca-
demic Press, (1980).
[6] S. Olariu, On source in comparability graphs, with applications, Dis-
crete Mathematics, pp. 289-292, (1992).
[7] A. Pnueli, A. Lempel, S. Even, Transitive orientation of graphs and
identification of permutation graphs, Canad. J. Math. 23, pp. 160–
175, (1971).
[8] W.T. Trotter, Combinatorics and parially ordered sets: dimension
theory, The Johns Hopkins University Press, (1992).
96 L. Alcon, N. Gudin˜o and M. Gutierrez
Liliana Alcon
Mathematics Department, Facultad Ciencias Exactas
Universidad Nacional de La Plata
C.C. 172 La Plata (1900) Argentina
liliana@mate.unlp.edu.ar
Noemi Gudin˜o
Mathematics Department, Facultad Ciencias Exactas
Universidad Nacional de La Plata
CONICET
C.C. 172 La Plata (1900) Argentina
noemigudino@mate.unlp.edu.ar
Marisa Gutierrez
Mathematics Department, Facultad Ciencias Exactas
Universidad Nacional de La Plata
CONICET
C.C. 172 La Plata (1900) Argentina
marisa@mate.unlp.edu.ar


End vertices in containment interval graphs

Servicio de Difusión de la Creación Intelectual

Matema´tica Contemporaˆnea, Vol. 45, 86–96c©2017, Sociedade Brasileira de Matema´ticaEnd vertices in containment intervalgraphsLiliana Alcon Noemi Gudin˜o Marisa GutierrezAbstractAn interval containment model of a graph maps vertices intointervals of a line in such a way that two vertices are adjacent ifand only if the corresponding intervals are comparable under theinclusion relation. Graphs admitting an interval containment modelare called containment interval graphs or CI graphs for short. Avertex v of a CI graph G is an end-vertex if there is an intervalcontainment model of G in which the left endpoint of the intervalcorresponding to v is less than all other endpoints. In this work,we present a characterization of end-vertices in terms of forbiddeninduced subgraphs.1 Introduction and previous resultsA graph G is a containment graph of intervals (or CI for short) if thereis a collection of intervals of the real line (Iw)w∈V (G) satisfying that uwis an edge of G if and only if Iu ⊂ Iw or Iw ⊂ Iu; the collection is calleda CI model of G. Without loss of generality, it can be assumed that theintervals of a CI model are closed, with positive length and no two have2000 AMS Subject Classification: 05C62, 05C75.Key Words and Phrases: comparability graphs, permutation graphs, containmentmodelsEnd vertices in containment interval graphs 871 2 2n1 2 2n 1 2 2n+ 11232n+ 12nn > 1 n > 2n > 2 n > 1J2n+1Figure 1: These graphs, together with the complement of the graphs inFigure 2, constitute a minimal family of forbidden induced subgraphs forcomparability graphs.a same endpoint [5, 8]. Let lw and rw denote, respectively, the left andthe right endpoint of the interval Iw. A vertex v is an end-vertex of G ifthere exists a CI model of G where lv < lw for every w ∈ V (G)− {v}.In this paper, we describe end-vertices of CI graphs using a self - com-plementary family of forbidden induced subgraphs. In addition, in Section3, homogeneously representable CI graphs (all vertices are end-vertices)are characterized by a simple finite family of forbidden induced subgraphs.A transitive orientation−→E of a graph G is an assignment of one of thetwo possible directions, −→uw or −→wu, to each edge uw ∈ E(G), such thatif −→uv ∈ −→E and −→vw ∈ −→E then −→uw ∈ −→E . Graphs admitting a transitiveorientation are called comparability graphs [5, 8].Next theorem, due to Gallai, provides a characterization of compara-bility graphs in terms of forbidden induced subgraphs. The complementof a graph G is the graph G =(V (G), E(G))such that uv ∈ E(G) if andonly if uv /∈ E(G).Theorem 1 ([3]). A graph G is a comparability graph if and only if Gcontains none of the graphs depicted in Figure 1, nor the complement ofthose in Figure 2, as induced subgraphs.A vertex v of a comparability graph G is a sink (respectively, a source)88 L. Alcon, N. Gudin˜o and M. Gutierrez1 2 n 123n− 1nn > 3n > 51 2 nn > 21 2 nn > 2T SG1MG2G3NnFigure 2: The complement of these graphs, together with the graphs inFigure 1, constitute a minimal family of forbidden induced subgraphs forcomparability graphs.End vertices in containment interval graphs 89if there exists a transitive orientation−→E of G such that −→vw /∈ −→E (−→wv /∈ −→E )for all w ∈ V (G). Olariu and Gimbel, almost simultaneously, obtaineda characterization of sinks in terms of forbidden induced subgraphs asfollows.Theorem 2 ([4, 6]). Let G be a comparability graph. A vertex v of G isa sink if and only if G contains none of the graphs A, B, C, D2k+1 withk ≥ 2, En with n ≥ 3, in Figure 3, with v as the designated vertex.CI graphs have been widely studied and characterized in different ways.For instance, they are co-comparability graphs, and in [1, 8] is showed therelationship between CI graphs and the partially ordered sets (or posets)of dimension at most 2. Given a poset P = (V,≤), the comparabilitygraph of P is GP = (V,E) with E = {uv : u < v or v < u in P}. Thedimension of a poset P is the minimal number of linear extension of Pwhose intersection is P [1]. The CI graphs are exactly the comparabilitygraphs of posets with dimension at most 2.Theorem 3 ([1]). A graph G is CI if and only if G and its complementG are comparability graphs.In addition, the class of CI graphs is equivalent to the class of permu-tation graphs [2, 7].2 Characterization of end-vertices of CI graphsIn the proof of the main Theorem 5, we will use the following two results.Remark 1 ([1]). If−→E is a transitive orientation of a CI graph G thenthere exists a CI model (Iw)w∈V (G) of G compatible with−→E ; ie. for eachpair of vertices u,w ∈ V (G), Iu ⊂ Iw if and only if −→uw ∈ −→E .Given a CI model of a graph G, the right set Rv of a vertex v ∈ V (G)is {w : lw < rv < rw}; and the left set Lv is {w : lw < lv < rw}. Clearly, ifw ∈ Rv ∩ Lv then Iv ⊂ Iw, and, consequently, w is adjacent to v.90 L. Alcon, N. Gudin˜o and M. GutierrezLemma 4. Let (Ivi)0≤i≤k be a CI model of a chordless path P = [v0v1 . . . vk]with k ≥ 2. If Iv1 ⊂ Iv0 then exactly one of the following statements holds:(i) v2 ∈ Rv0 and rv0 < rvi for 2 ≤ i ≤ k. (ii) v2 ∈ Lv0 and lvi < lv0 for2 ≤ i ≤ k.Proof. We proceed by induction. The proposition holds trivially whenk = 2. Let k > 2 and assume Iv1 ⊂ Iv0 . Since v2 is adjacent to v1 andnonadjacent to v0, we have that v2 ∈ Rv0 \ Lv0 or v2 ∈ Lv0 \Rv0 .Assume, w.l.g., that v2 ∈ Rv0 . Thus, rv0 < rvi for 2 ≤ i ≤ k − 1. Tocomplete the proof, we will show that rv0 < rvk .Since vk is adjacent to vk−1, it follows that Ivk−1 ⊂ Ivk or Ivk ⊂ Ivk−1 .In the former case, it is clear that rv0 < rvk−1 < rvk . Thus, assumeIvk ⊂ Ivk−1 ; and, in order to derive a contradiction, suppose that rvk < rv0 .Since rv0 < rvk−1 and vk−1 is not adjacent to v0, we have that lv0 < lvk−1 .Therefore lv0 < lvk−1 < lvk , and so Ivk ⊂ Iv0 , which contradicts the factthat vk and v0 are nonadjacent. Theorem 5. Let G be a connected CI graph. A vertex z is an end-vertexof G if and only if G contains none of the graphs A, A, B, B, C, C,D2k+1, D2k+1, E3, E2k, E2k, for k ≥ 2, as induced subgraphs with z asthe vertex highlighted in Figure 3.Proof. Let (Iw)w∈V (G) be a CI model of G with lz < lw for all w ∈V (G)− {z}. Let −→E be the orientation of G obtained by orienting u to v(−→uv) whenever Iu ⊂ Iv. Recall that two vertices are adjacent if and onlyif the interval corresponding to one of them is contained in the intervalcorresponding to the other. Clearly,−→E is a transitive orientation of G andz is a sink; thus, by Theorem 2, G does not contain the graphs A, B, C,D2k+1 with k ≥ 2 and En with n ≥ 3 as induced subgraph with z as thedesignated vertex. Observe that E3 = E3.Let H be any induced subgraph of G containing vertex z. Denote byH ′ the graph H plus a vertex z′ adjacent to z, that is, H ′ = (V (H) ∪z′, E(H) ∪ z′z) and denote by H ′′ the graph H ′ plus a vertex z′′ adjacentEnd vertices in containment interval graphs 91A A¯B B¯C C¯DnEn E¯n1 2 3 nx123nx1 2 3 nxy123nnD¯nFigure 3: The highlighted vertices in the graphs A, B, C, D2k+1 withk ≥ 2, En with n ≥ 3, cannot be a sink of a comparability graph. Thehighlighted vertices in the graphs A, A, B, B, C, C, D2k+1, D2k+1, E3,E2k, E2k, for k ≥ 2, cannot be an end-vertex of a CI graph.92 L. Alcon, N. Gudin˜o and M. Gutierrezto z′, that is, H ′′ = (V (H ′′) ∪ z′′, E(H) ∪ z′′z′). We claim that H ′ andH ′′ are CI graphs .Indeed, CI models ofH ′ andH ′′ can be obtained by adding to (Iw)w∈V (H)the intervals Iz′ = [lz + , lz + 2] and Iz′′ = [lz − , lz + 3] with  smallenough.Now, in order to derive a contradiction, assume that H is any one ofthe graphs A, B, C, D2k+1, E2k, with k ≥ 2, and z is the vertex ofH highlighted in Figure 3. Notice that (A)′′ = (B)′ = T in Figure 2;(C)′ = M in Figure 2; (D2k+1)′ = J2k+1 in Figure 1 and E2k ′ = N2k+1 inFigure 2.Thus, by the previous claim (2), these graphs are CI, which contradictsthe fact that neither the graphs depicted in Figures 1 and 2 nor theircomplements are CI graphs.To prove the converse, notice first that, by Theorem 2, the vertex zmust be a sink of G, which means there exists a transitive orientation−→Eof G in which all edges incident in z are oriented towards z.Let G′ be the graph G plus a pendent vertex z′ adjacent to z. Since atransitive orientation−→E′ of G′ can be obtained adding−→z′z to−→E , it followsthat G′ is a comparability graph and z is a sink of G′.Assume, in order to derive a contradiction, that G′ is not CI. Thus,G′ contains an induced subgraph H which is either a graph in Figure 2 orthe complement of a graph in Figure 1; even more, H must contain z′, soH must have a vertex of degree one and z must be the only neighbor ofthat vertex.An inspection of those figures reveals that H has to be one of the graphsT , S, M , G1, G2, G3 or Nn for some n ≥ 4, in Figure 2; or the complementof the graph J2k+1 for some k ≥ 2 in Figure 1. In the following paragraph,we will show that each case implies a contradiction, therefore G′ is a CIgraph.* If H is any one of the graphs S, G1, G2 or G3, then G contains thegraph A in Figure 3 with z as the highlighted vertex.* If H is the graph T , then G contains the graph B in Figure 3 with zEnd vertices in containment interval graphs 93as the highlighted vertex.* If H is the graph M , then we have to consider two cases depending onwhich vertex of degree one is z′: its neighbor z is adjacent to two verticeswith degree 3, or its neighbor z is adjacent to a vertex with degree 3 andother with degree 2. In the first case, G contains the graph A in Figure 3with z as the highlighted vertex; and, in the latter, G contains the graphC in Figure 3 with z as the highlighted vertex.* If H is the graph N4, then G contains the graph E3 in Figure 3 withz as the highlighted vertex.* If H is the graph Nn for some n > 4, then, again, we consider twocases depending on which vertex of degree one is z′: if z′ is the vertexlabelled 1 (or n) then G contains the graphs En−1 and Dn−1 in Figure 3with z as the highlighted vertex. Notice that En−1 is forbidden when nis odd and Dn−1 is forbidden when n is even. If z′ is neither the vertexlabelled 1 nor the vertex labelled n, then G contains the graph A in Figure3 with z as the highlighted vertex.Thus, we have proved that G′ is CI. By Remark 1, there exist a CImodel (Iw)w∈V (G′) compatible with−→E′, i.e. −→uv ∈ −→E′ implies Iu ⊂ Iv.Clearly, if Iz is an end interval in this model, the vertex z is an end-vertex of G, and the proof follows.So, we assume that the left set Lz and the right set Rz of z are nonempty. The fact that z is the only neighbor of z′ implies Lz ∩ Rz = ∅.Let x be a vertex of Lz minimizing the distance to z in G′; and Pzx =[v0 = z, v1, v2, ...vk−1, vk = x] be a shortest path joining z with x. Since zis a sink, we have that −→v1z ∈−→E′, which implies that Iv1 ⊂ Iz. Therefore,by Lemma 4, v2 ∈ Rz and rz < rvi for every i, or v2 ∈ Lz and lvi < lz forevery i. Since vk = x ∈ Lz, the former is not possible; thus, v2 ∈ Lz whichimplies k = 2 and the existence of the induced path [z, v1, x]. Analogously,there exist vertices v′1 and y ∈ Rz inducing the path [z, v′1, y].The proof will be complete showing that the concatenation of bothpaths induces the graph A with z as the highlighted vertex.Clearly, x and y are non-adjacent. We claim that x is non-adjacent to94 L. Alcon, N. Gudin˜o and M. Gutierrezv′1. Indeed, if x is adjacent to v′1, we have that rv′1 < rx; on the other hand,since z′ has degree 1 and Iz′ ⊂ Iz, it follows that rx < rz′ and lz′ < ly.Therefore, lz′ < ly < lv′1 < rv′1 < rx < rz′ which implies Iv′1 ⊂ Iz′ , contraryto the fact that z′ is adjacent only to z. Analogously, y is non-adjacent tov1. As a consequence of Theorems 2 and 5, we have the following result.Corollary 6. Let G be a CI graph. A vertex z is an end-vertex of G ifand only if z is a sink of G and G.3 Characterization of CI homogeneously repre-sentable graphsA CI graph G is homogeneously representable if every one of its verticesis an end-vertex.Theorem 7. Let G be CI. The graph G is homogeneously representableif and only if it contains none of the graphs A, A, C, C, E3, depicted inFigure 3, as induced subgraphs.Proof. It is a straightforward consequence of Theorem 5 and the facts: Ais an induced subgraph of B and of D2k+1, for k ≥ 2; and E3 is an inducedsubgraph of E2k, for k ≥ 2. In the previous theorem, the condition that G is a CI graph can berelaxed by adding C5 to the family of forbidden induced subgraph.Theorem 8. A graph G is CI homogeneously representable if and only ifit contains none of the graphs C5, A, A, C, C, E3, depicted in Figure 3,as induced subgraphs.Proof. The direct implication follows from Theorem 7 and the fact thatC5 is not a comparability graph.To prove the converse, keep in mind that the graphs depicted in Figures1 and 2 and their complements are a family of forbidden induced subgraphsof a CI graph; and notice that C5 and E3 are self-complementary. End vertices in containment interval graphs 95References[1] B. Dushnik, E. Miller, Partially ordered sets, Amer. J. Math. 63, pp.600–610, (1941).[2] S. Even, A. Pnueli, A. Lempel Permutation graphs and transitivegraphs, J. ACM 19, pp. 400–410, (1972).[3] T. Gallai, Transitiv Orientierbare Graphen, Acta. Math. Hungar. 18,pp. 25–66, (1967).[4] J. Gimbel, Source in posets and comparability graphs, Order 9, pp.361–365,(1992).[5] M.C. Golumbic, Algorithmic Graph Theory and Perfect Graphs, Aca-demic Press, (1980).[6] S. Olariu, On source in comparability graphs, with applications, Dis-crete Mathematics, pp. 289-292, (1992).[7] A. Pnueli, A. Lempel, S. Even, Transitive orientation of graphs andidentification of permutation graphs, Canad. J. Math. 23, pp. 160–175, (1971).[8] W.T. Trotter, Combinatorics and parially ordered sets: dimensiontheory, The Johns Hopkins University Press, (1992).96 L. Alcon, N. Gudin˜o and M. GutierrezLiliana AlconMathematics Department, Facultad Ciencias ExactasUniversidad Nacional de La PlataC.C. 172 La Plata (1900) Argentinaliliana@mate.unlp.edu.arNoemi Gudin˜oMathematics Department, Facultad Ciencias ExactasUniversidad Nacional de La PlataCONICETC.C. 172 La Plata (1900) Argentinanoemigudino@mate.unlp.edu.arMarisa GutierrezMathematics Department, Facultad Ciencias ExactasUniversidad Nacional de La PlataCONICETC.C. 172 La Plata (1900) Argentinamarisa@mate.unlp.edu.ar

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End vertices in containment interval graphs

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