Cellular Networks Modeled by Distance Hereditary Graphs Are Maximum-Clique Perfect

Cellular Networks Modeled by Distance Hereditary Graphs Are

Maximum-Clique Perfect

*

Chuan-Min Lee

Department of Computer and Communication Engineering

Ming Chuan University

5 De Ming Rd., Guishan District, Taoyuan County 333, Taiwan.

[email protected]

Abstract

_{-In modern cellular telecommunications} systems, the entire service area of a country is divided into cells. Cells are normally thought of as hexagonal grids. One common method used to place transmitters for cellular telephones is to place them at the corner points of each hexagonal grid. Motivated by the placement of transmitters for cellular telephones, Chang, Kloks, and Lee introduced the concept of maximum-clique transversal sets on graphs in 2001. In this paper, we show that cellular networks modeled by distance-hereditary graphs are maximum-clique perfect. The maximum-clique transversal number and the maximum-clique independence number of a distance hereditary graph can be computed in linear time.

Keywords: Algorithm, Maximum-Clique

Transversal Set, Maximum-Clique Independent Set, Distance-Hereditary Graph..

1. Introduction

All graphs in this paper are undirected, finite, and simple. Let G = (V, E) be a graph with |V| = n and |E| = m. For a graph G, we also use V(G) and

E(G) to denote the vertex set and edge set of G, respectively. We use G[W] to denote a subgraph of G induced by a subset W of V. For any vertex in

v

∈

V

, a clique is a subset of pairwise adjacent vertices of V. A maximal clique is a clique that is not a proper subset of any other clique. A clique is maximum if there is no clique of G of larger cardinality. The clique number of G, denoted by

w(G), is the cardinality of a maximum clique of G. We use Q(G) to denote the collection of all maximum cliques of G. A maximum-clique

transversal set of a graph G = (V, E) is a subset of

V intersecting all maximum cliques of G. The

maximum-clique transversal number of G, denoted byτ_M(G) , is the minimum cardinality of a maximum-clique transversal set of G. The

maximum-clique transversal set problem is to find a maximum-clique transversal set of G of

minimum cardinality. A maximum-clique

independent set of G is a collection of pairwise

disjoint maximum cliques of G. The

maximum-clique independence number of G, denoted by αM(G), is the maximum cardinality of a maximum-clique independent set of G. The

maximum-clique independent set problem is to find a maximum-clique independent set of G of maximum cardinality. It is clear that the weak

duality inequalityα_M(G)≤τ_M(G)holds for any graph G.

Clique transversal and clique independent sets are closely related to maximum-clique transversal and maximum-clique independent sets. They have been studied in [3,4,5,6,7]. In this paper, we define a graph G to be maximum-clique perfect if

) ( )

(H M H

M α

τ = for every induced subgraph H of

G.

A graph G = (V,E) is called distance

hereditary if every pair of vertices are equidistant in every connected induced subgraph containing them. The following theorem shows that

distance-hereditary graphs can be defined

recursively.

Theorem 1. [2] Distance-hereditary graphs can be

defined recursively as follows:

1. A graph consisting of only one vertex is distance hereditary, and the twin set is the vertex itself.

2. If G1 and G2 are disjoint distance hereditary graphs with the twin sets TS(G1)

and TS(G2) , respectively, then the graph

2 1 G

G

G= ∪ is a distance-hereditary graph

and the twin set of G is TS(G) = TS(G1)

∪

*This research was partially supported by National Science

Council in Taiwan, under the grant number NSC-97-2218-E-130-002-MY2.

TS(G2). G is said to be obtained from G1 and

G2 by a false twin operation.

3. If G1 and G2 are disjoint distance hereditary graphs with the twin sets TS(G1)

and TS(G2) , respectively, then the graph

G obtained by connecting every vertex of

TS(G1) to all vertices of TS(G2) is a distance hereditary graph, and the twin set of

G is TS(G) = TS(G1)

∪

TS(G2). G is said to be obtained from G1 and G2 by a true twin operation.

4. If G1 and G2 are disjoint distance hereditary graphs with the twin sets TS(G1)

and TS(G2) , respectively, then the graph

G obtained by connecting every vertex of

TS(G1) to all vertices of TS(G2) is a distance hereditary graph, and the twin set of

G is TS(G) = TS(G1). G is said to be obtained from G1 and G2 by a pendant vertex operation. Following Theorem 1, a binary ordered decomposition tree can be obtained in linear time [1]. In this decomposition tree, each leaf is a single vertex graph, and each internal node represents one of the three operations: pendant vertex operation (labeled by P), true twin operation (labeled by T), and false twin operation (labeled by F). This ordered decomposition tree is called a

PTF-tree.

2. Main Result

In this section, we will prove that distance hereditary graphs are maximum-clique perfect. Due to space limitations, we have to omit the proof of each lemma and theorem in this section.

Definition 1. Recall that Q(G) denotes the

collection of all maximum cliques of G. Hence

Q(G[TS(G)]) is the collection of all maximum cliques of G[TS(G)] . We use QTS(G) to

denote the collection of all maximum cliques of G which are maximum cliques of G[TS(G)] and

use Q (G)

TS to denote the collection of all

maximum cliques of G which are not maximum

cliques of G[TS(G)]. Hence Q(G) =

QTS(G) Q (G)

TS

∪ . Let QE(G) = Q(G)

∪

Q(G[TS(G)]) . QE(G) denotes the collection of

all maximum cliques of G and all maximum cliques of G[TS(G)] .

Definition 2. Suppose that a distance hereditary graph G is obtained from two disjoint distance hereditary graphs G1 and G2 by one of the three operations: pendant vertex operation, true twin

operation, and false twin operation. We use w, wt,

w1,

w

t₁, w2, and w to denote the clique numbers t₂

of G , G[TS(G)] , G1 , G1[TS(G1)] , G2 , and G2[TS(G2)] , respectively.

Lemma 1. Suppose that G is a graph obtained

from two disjoint distance hereditary graphs G1 and G2 by a false twin operation. Let

i

∈

{

1

,

2

}

. Then, (1)        = ∪ > > = . if )]) ( [ ( )]) ( [ ( ; if )]) ( [ ( ; if )]) ( [ ( )] ( [ ( 2 1 1 2 2 1 2 2 1 1 2 2 1 1 t t t t t t w w G TS G Q G TS G Q w w G TS G Q w w G TS G Q G TS G Q (2)      = ∪ > > = . if ) ( ) ( ; if ) ( ; if ) ( ) ( 2 1 2 1 1 2 2 2 1 1 w w G Q G Q w w G Q w w G Q G Q (3)      = ∪ > > = . if ) ( ) ( ; if ) ( ; if ) ( ) ( 2 1 2 1 1 2 2 2 1 1 w w G Q G Q w w G Q w w G Q G Q TS TS TS TS TS (4)       = ∪ > > = . if ) ( ) ( ; if ) ( ; if ) ( ) ( 2 1 2 1 1 2 2 2 1 1 w w G Q G Q w w G Q w w G Q G Q TS TS TS TS TS (5)                 > > < > ∪ = > ∪ < = ∪ = = ∪ = − − − − − − − − − − ; and if ) ( ; and if ) ( )]) ( [ ( ; and if ) ( ) ( ; and if )]) ( [ ( ) ( ; and if ) ( ) ( ) ( 3 3 3 2 1 3 3 3 3 2 1 2 1 3 3 3 2 1 2 1 i i t t i E i i t t i TS i i t t i TS i E i i t t i i i E t t E E E w w w w G Q w w w w G Q G TS G Q w w w w G Q G Q w w w w G TS G Q G Q w w w w G Q G Q G Q i i i i i i

Definition 3. Suppose that G is a graph obtained

from two disjoint distance hereditary graphs G1 and G2 by a true twin operation or a pendant vertex

operation. We use Q12(G) to denote

)])}. ( [ ( and )]) ( [ ( | {q1∪q2 q1∈QG1TSG1 q2∈QG2TSG2

Lemma 2. Suppose that G is a graph obtained

from two disjoint distance hereditary graph G1 and

G2 by a true twin operation. Then, (1) Q(G[TS(G)]) = Q12(G).

(2)                    = < + ∪ > < + = > < + = = = + ∪ ∪ > = + ∪ > = + ∪ > + = . if ) ( ) ( ; and if ) ( ) ( ; and if ) ( ) ( ; if ) ( ) ( ) ( ; if ) ( ) ( ; if ) ( ) ( }; , max{ if ) ( ) ( 2 1 2 1 1 2 2 2 2 2 1 1 1 1 2 1 2 1 12 1 2 2 12 2 1 1 12 2 1 12 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G Q G Q w w w w w G Q G Q w w w w w G Q G Q w w w w G Q G Q G Q w w w w G Q G Q w w w w G Q G Q w w w w G Q G Q t t TS TS t t TS t t TS t t TS TS t t TS t t TS t t (3)                    = < + ∪ > < + = > < + = = = + ∪ > = + > = + > + = . if ) ( ) ( ; and if ) ( ) ( ; and if ) ( ) ( ; if ) ( ) ( ; if ) ( ; if ) ( }; , max{ if ) ( 2 1 2 1 1 2 2 2 2 2 1 1 1 1 2 1 2 1 1 2 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G Q G Q w w w w w G Q G Q w w w w w G Q G Q w w w w G Q G Q w w w w G Q w w w w G Q w w w w G Q t t TS TS t t TS t t TS t t TS TS t t TS t t TS t t TS φ (4)    + ≥ = otherwise. , }; , max{ if )]) ( [ ( ) ( 1 2 1 2 φ w w w w G TS G Q G Q_TS t t (5)    ∪ ≥ + = otherwise. , ) ( )]) ( [ ( }; , max{ if ) ( ) ( 1 2 1 2 G Q G TS G Q w w w w G Q G Q_E t t

Lemma 3. Suppose that G is a graph obtained

from two disjoint distance hereditary graph G1 and

G2 by a true twin operation. Then,

(1)                      = < + ∪ > < + = > < + = = = + ∪ ∪ > = + ∪ > = + ∪ > + = . if ) ( ) ( ; and if ) ( ) ( ; and if ) ( ) ( ; if ) ( ) ( ) ( ; if ) ( ) ( ; if ) ( ) ( }; , max{ if ) ( ) ( 2 1 2 1 1 2 2 2 2 2 1 1 1 1 2 1 2 1 12 1 2 2 12 2 1 1 12 2 1 12 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G Q G Q w w w w w G Q G Q w w w w w G Q G Q w w w w G Q G Q G Q w w w w G Q G Q w w w w G Q G Q w w w w G Q G Q t t TS TS t t TS t t TS t t TS TS t t TS t t TS t t (2)                        = < + ∪ > < + ∪ > < + = = + ∪ ∪ > = + ∪ ∪ > = + ∪ > + ∪ = . if ) ( ) ( ; and if ) ( )]) ( [ ( ; and if ) ( ; if ) ( ) ( ) ( ; if ) ( ) ( )]) ( [ ( ; if ) ( ) ( }; , max{ if )]) ( [ ( ) ( ) ( 2 1 2 1 1 2 2 2 1 1 2 1 1 1 2 1 2 1 12 1 2 2 12 1 1 2 1 1 12 2 1 1 1 12 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G Q G Q w w w w w G Q G TS G Q w w w w w G Q w w w w G Q G Q G Q w w w w G Q G Q G TS G Q w w w w G Q G Q w w w w G TS G Q G Q G Q t t TS E t t TS t t E t t TS E t t TS t t E t t E (3) Q(G[TS(G)])=Q(G1[TS(G1)]) , QTS(G)=φ , and ) ( ) (G QG Q_TS = .

Lemma 4. Suppose that G is a graph obtained

from two disjoint distance hereditary graphs G1 and G2 by a true twin operation or a pendant vertex operation. Let S be a maximum-clique transversal

set of G . If _{max{ , }} 2 1 2 1 w w w wt + t ≥ , then either ) (G1 TS

S∩ is a maximum-clique transversal set of

G1[TS(G1)] orS∩TS(G2) is a maximum-clique transversal set of G2[TS(G2)].

Definition 4. A strong maximum-clique transversal set of G is a subset of V that intersects all cliques in QE(G) . We use SCT(G) to represent a strong maximum-clique transversal set of G .

Definition 5. A weak maximum-clique transversal set of G is a subset of V that intersects all cliques in Q_TS(G). We use WCT(G) to represent a weak

maximum-clique transversal set of G.

Definition 6. An expanded maximum-clique independent set of G is a collection of pairwise disjoint cliques in QE(G). We use ECI(G) to

represent an expanded maximum-clique

independent set of G .

Definition 7. A weak maximum-clique

independent set of G is a collection of pairwise disjoint cliques inQ_TS(G). We use WCI(G) to

represent a weak maximum-clique independent set of G .

maximum-clique transversal set and a maximum-clique independent set of G , respectively. We say that a distance hereditary graph G holds the strong duality if there exist a

CT(G), a CI(G), a CT(G[TS(G)]) , a

CI(G[TS(G)]) , a WCT(G) , a WCI(G) , an

SCT(G) , and an ECI(G) such that the following five conditions are satisfied: (1) |CT(G) | = |CI(G)|, (2) |CT(G[TS(G)])| = |CI(G[TS(G)])|, (3) |WCT(G)| = |WCI(G)| , (4) |SCT(G)| = |ECI(G)| , and (5) WCI(G) is a subset of

ECI(G) . Let XI(G) denote ECI(G)−WCI(G).

Definition 9. Assume that G is a distance

hereditary graph formed from two disjoint distance hereditary graphs G1 and G2 by a pendant vertex operation or by a true twin operation, and both G1 and G2 hold the strong duality. Suppose that XI(G1)

={ , , } 1 1 ck c K , XI(G2) ={ , , } 2 1 dk d K , CI(G1[TS(G1)]) = { , } 1 1 pr p K , and CI(G1[TS(G1)]) = { , } 2 1 qr q K . We

have the following definitions.

(1) Let k = min{k1, k2}. We let XX(G) = } 1 | {ci∪di ≤i≤k and let XX’(G) = } , , { 1 1 k k c c K + if k1 > k and XX’(G) =

φ

otherwise.

(2) Let r = min{r1, r2}. We let TT(G) = } 1 | {p_i∪q_i ≤i≤r and let TT’(G) = } , , {pr+1K pr₁ if r1 > r and TT’(G) =

φ

otherwise.

(3) Let

l

_{= min{k1, r2}. We let XT(G) =}

} 1 | {_{c ∪q ≤i≤}l i i and let XT’(G) = } , , { 1 1 ck c_l+ K if k1 >

l

and XT’(G) =

φ

otherwise.

(4) Let s = min{r1, k2}. We let TX(G) = } 1 | {p_i∪d_i ≤i≤s and let TX’(G) = } , , {ps+1K pr₁ if r1 > s and TX’(G) =

φ

otherwise.

Lemma 5. Assume that G is a graph of single

vertex and v is the vertex of G. There exist the following sets: (1) CT(G)={v}, (2) CT(G[TS(G)])=

{v}, (3) SCT(G)={v}, (4) WCT(G)=

φ

, (5) WCI(G)

=

φ

, (6) ECI(G) = {v}}, (7) CI(G[TS(G)]) = {{v}}, and (8) CI(G) = {{v}} such that G holds the strong duality.

Lemma 6. Assume that G is obtained from two

disjoint distance hereditary graphs G1 and G2 by a false twin operation , and both G1 and G2 hold the

strong duality. Let

i

∈

{

1

,

2

}

. There exist the following sets such that G holds the strong duality. (1)      = ∪ > > = . if ) ( ) ( ; if ) ( ; if ) ( ) ( 2 1 2 1 1 2 2 2 1 1 w w G CT G CT w w G CT w w G CT G CT (2)        = ∪ > > = . if )]) ( [ ( )]) ( [ ( ; if )]) ( [ ( ; if )]) ( [ ( )]) ( [ ( 1 2 1 2 2 1 2 2 1 1 2 2 1 1 t t t t t t w w G TS G CT G TS G CT w w G TS G CT w w G TS G CT G TS G CT (3)                  > > < > ∪ = > ∪ > = ∪ = = ∪ = − − − − − − − − − − ; and if ) ( ; and if ) ( )]) ( [ ( ; and if ) ( ) ( ; and if )]) ( [ ( ) ( ; and if ) ( ) ( ) ( 3 3 3 2 1 3 3 3 3 2 1 2 1 3 3 3 2 1 2 1 i i t t i i i t t i i i t t i i i i t t i i i t t w w w w G SCT w w w w G WCT G TS G CT w w w w G WCT G SCT w w w w G TS G CT G SCT w w w w G SCT G SCT G SCT i i i i i i (4)      = ∪ > > = . if ) ( ) ( ; if ) ( ; if ) ( ) ( 2 1 2 1 1 2 2 2 1 1 w w G WCT G WCT w w G WCT w w G WCT G WCT (5)      = ∪ > > = . if ) ( ) ( ; if ) ( ; if ) ( ) ( 2 1 2 1 1 2 2 2 1 1 w w G WCI G WCI w w G WCI w w G WCI G WCI (6)                  > > < > ∪ = > ∪ > = ∪ = = ∪ = − − − − − − − − − − ; and if ) ( ; and if ) ( )]) ( [ ( ; and if ) ( ) ( ; and if )]) ( [ ( ) ( ; and if ) ( ) ( ) ( 3 3 3 2 1 3 3 3 3 2 1 2 1 3 3 3 2 1 2 1 i i t t i i i t t i i i t t i i i i t t i i i t t w w w w G ECI w w w w G WCI G TS G CI w w w w G WCI G ECI w w w w G TS G CI G ECI w w w w G ECI G ECI G ECI i i i i i i (7)        = ∪ > > = . if )]) ( [ ( )]) ( [ ( ; if )]) ( [ ( ; if )]) ( [ ( )]) ( [ ( 1 2 1 2 2 1 2 2 1 1 2 2 1 1 t t t t t t w w G TS G CI G TS G CI w w G TS G CI w w G TS G CI G TS G CI (8)      = ∪ > > = . if ) ( ) ( ; if ) ( ; if ) ( ) ( 2 1 2 1 1 2 2 2 1 1 w w G CI G CI w w G CI w w G CI G CI

Lemma 7. Assume that G is obtained from two

disjoint distance hereditary graphs G1 and G2 by a pendant vertex operation , and both G1 and G2 hold the strong duality.

                         = < + ∪ > < + = > < + = = = + ∪ ∪ > = + ∪ > = + ∪ > + = . if ) ( ) ( ; and if ) ( ) ( ; and if ) ( ) ( ; if )} ( ) ( ), ( ) ( min{ ; if )])} ( [ ( ) ( ), ( min{ ; if )])} ( [ ( ) ( ), ( min{ }; , max{ if )])} ( [ ( )]), ( [ ( min{ ) ( ) 1 ( 2 1 2 1 1 2 2 2 2 2 1 1 1 1 2 1 2 1 2 1 1 2 1 1 2 2 2 1 2 2 1 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G WCT G WCT w w w w w G WCT G CT w w w w w G WCT G CT w w w w G SCT G WCT G WCT G SCT w w w w G TS G CT G WCT G SCT w w w w G TS G CT G WCT G SCT w w w w G TS G CT G TS G CT G CT t t t t t t t t t t t t t t (2) CT(G[TS(G)])=CT(G1[TS(G1)]). (3) WCT(G) = CT(G).                        = < + ∪ > < + ∪ > < + = = + ∪ > = + ∪ > = + > + = . if ) ( ) ( ; and if )]) ( [ ( ) ( ; and if ) ( ; if ) ( ) ( ; if )]) ( [ ( ) ( ; if ) ( }; , max{ if )]) ( [ ( ) ( ) 4 ( 2 1 2 1 1 2 2 1 1 2 2 1 1 1 2 1 2 1 1 2 1 1 2 2 1 1 2 1 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G WCT G SCT w w w w w G TS G CT G WCT w w w w w G SCT w w w w G WCT G SCT w w w w G TS G CT G WCT w w w w G SCT w w w w G TS G CT G SCT t t t t t t t t t t t t t t                        = < + ∪ > < + > < + = = + ∪ ∪ > = + ∪ > = + ∪ > + = . if ) ( ) ( ; and if ) ( ; and if ) ( ; if ) ( ) ( ) ( ; if ) ( ) ( ; if ) ( ) ( }; , max{ if ) ( ) ( ) 5 ( 2 1 2 1 1 2 2 2 2 1 1 1 2 1 2 1 1 2 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G WCI G WCI w w w w w G WCI w w w w w G WCI w w w w G XX G WCI G WCI w w w w G TX G WCI w w w w G XT G WCI w w w w G TT G WCI t t t t t t t t t t t t t t                          = < + ∪ > < + ∪ > < + = = + ′ ∪ ∪ ∪ > = + ′ ∪ ∪ > = + ′ ∪ ∪ > + ′ ∪ = . if ) ( ) ( ; and if )]) ( [ ( ) ( ; and if ) ( ; if ) ( ) ( ) ( ) ( ; if ) ( ) ( ) ( ; if ) ( ) ( ) ( }; , max{ if ) ( ) ( ) ( ) 6 ( 2 1 2 1 1 2 2 1 1 2 2 1 1 1 2 1 2 1 1 2 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G WCI G ECI w w w w w G TS G CI G WCI w w w w w G ECI w w w w G X X G XX G WCI G WCI w w w w G X T G TX G WCI w w w w G T X G XT G WCI w w w w G T T G TT G ECI t t t t t t t t t t t t t t (7) CI(G[TS(G)])=CI(G₁[TS(G₁)]). (8) CI(G) = WCI(G).

Lemma 8. Assume that G is obtained from two

disjoint distance hereditary graphs G1 and G2 by a true twin operation , and both G1 and G2 hold the strong duality.                              = < + ∪ > < + = > < + = = = + ∪ ∪ > = + ∪ > = + ∪ > + = . if ) ( ) ( ; and if ) ( ) ( ; and if ) ( ) ( ; if )} ( ) ( ), ( ) ( min{ ; if )])} ( [ ( ) ( ), ( min{ ; if )])} ( [ ( ) ( ), ( min{ }; , max{ if )])} ( [ ( )]), ( [ ( min{ ) ( ) 1 ( 2 1 2 1 1 2 2 2 2 2 1 1 1 1 2 1 2 1 2 1 1 2 1 1 2 2 2 1 2 2 1 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G WCT G WCT w w w w w G WCT G CT w w w w w G WCT G CT w w w w G SCT G WCT G WCT G SCT w w w w G TS G CT G WCT G SCT w w w w G TS G CT G WCT G SCT w w w w G TS G CT G TS G CT G CT t t t t t t t t t t t t t t )]). ( [ ( )]), ( [ ( min{ )]) ( [ ( ) 2 ( 2 2 1 1 G TS G CT G TS G CT G TS G CT =

                       = < + ∪ > < + > < + = = + ∪ > = + > = + > + = . if ) ( ) ( ; and if ) ( ; and if ) ( ; if ) ( ) ( ; if ) ( ; if ) ( }; , max{ if ) ( ) 3 ( 2 1 2 1 1 2 2 2 2 1 1 1 2 1 2 1 1 2 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G WCT G WCT w w w w w G WCT w w w w w G WCT w w w w G WCT G WCT w w w w G WCT w w w w G WCT w w w w G WCT t t t t t t t t t t t t t t φ                                  = < + ∪ ∪ > < + ∪ > < + ∪ = = + ∪ ∪ > = + ∪ > = + ∪ > + = . if )} ( ) ( ), ( ) ( min{ ; and if )])} ( [ ( ) ( ), ( min{ ; and if )])} ( [ ( ) ( ), ( min{ ; if )} ( ) ( ), ( ) ( min{ ; if )])} ( [ ( ) ( ), ( min{ ; if )])} ( [ ( ) ( ), ( min{ }; , max{ if )])} ( [ ( )]), ( [ ( min{ ) ( ) 4 ( 2 1 2 1 2 1 1 2 2 1 1 2 2 2 1 1 2 2 1 1 2 1 2 1 2 1 1 2 1 1 2 2 2 1 2 2 1 1 2 1 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G SCT G WCT G WCT G SCT w w w w w G TS G CT G WCT G SCT w w w w w G TS G CT G WCT G SCT w w w w G SCT G WCT G WCT G SCT w w w w G TS G CT G WCT G SCT w w w w G TS G CT G WCT G SCT w w w w G TS G CT G TS G CT G SCT t t t t t t t t t t t t t t                        = < + ∪ > < + > < + = = + ∪ > = + > = + > + = . if ) ( ) ( ; and if ) ( ; and if ) ( ; if ) ( ) ( ; if ) ( ; if ) ( }; , max{ if ) ( ) 5 ( 2 1 2 1 1 2 2 2 2 1 1 1 2 1 2 1 1 2 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G WCI G WCI w w w w w G WCI w w w w w G WCI w w w w G WCI G WCI w w w w G WCI w w w w G WCI w w w w G WCI t t t t t t t t t t t t t t φ                        = < + ∪ ∪ > < + ∪ > < + ∪ = = + ∪ ∪ > = + ∪ > = + ∪ > + = . if ) ( ) ( ) ( ; and if ) ( ) ( ; and if ) ( ) ( ; if ) ( ) ( ) ( ; if ) ( ) ( ; if ) ( ) ( }; , max{ if ) ( ) ( ) 6 ( 2 1 2 1 1 2 2 2 2 1 1 1 2 1 2 1 1 2 2 2 1 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 2 1 w w w w G XX G WCI G WCI w w w w w G TX G WCI w w w w w G XT G WCI w w w w G XX G WCI G WCI w w w w G TX G WCI w w w w G XT G WCI w w w w G TT G ECI t t t t t t t t t t t t t t (7) CI(G[TS(G)]) = TT(G). (8)    + ≥ = otherwise. , ) ( }; , max{ if ) ( ) ( 1 2 1 2 G WCI w w w w G ECI G CI t t

Theorem 2. Distance hereditary graphs are

maximum-clique perfect.

Theorem 3. For any distance hereditary graph G,

) ( and ) (G _M G M

α

τ

can be computed in linear

time.

References

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