The clique-separator graph for chordal graphs

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Contents lists available atScienceDirect

Discrete Applied Mathematics

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

The clique-separator graph for chordal graphs

I

Louis Ibarra

College of Computing and Digital Media, DePaul University, 243 S. Wabash Ave., Chicago, IL 60604, USA

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

Received 28 December 2007 Accepted 8 February 2009 Available online 5 April 2009 Keywords: Chordal graph Clique tree Interval graph Split graph a b s t r a c t

We present a new representation of a chordal graph called theclique-separator graph, whose nodes are the maximal cliques and minimal vertex separators of the graph. We present structural properties of the clique-separator graph and additional properties when the chordal graph is an interval graph, proper interval graph, or split graph. We also characterize proper interval graphs and split graphs in terms of the clique-separator graph. We present an algorithm that constructs the clique-separator graph of a chordal graph in

O(n3)time and of an interval graph inO(n2)time, wherenis the number of vertices in the graph.

©2009 Elsevier B.V. All rights reserved.

1. Introduction

Chordal graphs and clique trees have an extensive literature and applications in areas such as computational biology, databases, sparse matrix computation, and statistics [2,12,20]. In a clique treeTof a chordal graphG, the nodes ofTare the maximal cliques ofGand the edges ofT correspond to the minimal vertex separators ofG. In this paper, we introduce the clique-separator graphGof a chordal graphG. The graphGis a mixed graph where the nodes are the maximal cliques and minimal vertex separators ofGand the (directed) arcs and (undirected) edges represent the containment relations between the maximal cliques and minimal vertex separators ofG. The clique-separator graphGis unique and its structure reflects the structure ofG. We show thatGhas various structural properties including (a) the edges ofGinduce a forest, i.e., deleting the arcs ofGproduces a forest, (b) contracting every tree of this forest into a single node yields a directed acyclic graph, (c) for every minimal vertex separatorSofG, the vertices ofGseparated bySare specified by the nodes ofGseparated by the node corresponding toS.

WhenGis an interval graph, the clique-separator graphGhas additional structural properties including (d) every node inGhas at most two predecessors and (e) every tree of the forest (with certain leaves removed) is a path that is the unique clique path on those nodes. WhenGis a proper interval graph, the clique-separator graphGhas exactly one tree, which is a clique path, and no arcs. WhenGis a split graph, the clique-separator graphGhas a simple structure centered around one node ofG. (Recall that {proper interval graphs}

{interval graphs}

{chordal graphs} and that {split graphs}

{chordal graphs}.)

The clique tree of a chordal graphGdoes not appear to have structural properties whenGbelongs to a subclass of chordal graphs. For example, the starK1,nis an interval graph, a split graph, and a ptolemaic graph, and yet any tree on its maximal cliques is a clique tree ofK1,n. The clique tree is not unique and very different chordal graphs can have isomorphic clique trees. For example, the path onn

+

1 vertices has exactly one clique tree, which is isomorphic to a clique tree ofK1,n, which has more than exponentially many clique trees.

The clique-separator graph is different from theclique graphin [24], theweighted clique intersection graphin [1,2], and the clique graphin [10]. The graph in [24] is the intersection graph of the set of maximal cliques of a chordal graphG, i.e., the nodes

I Some of these results were part of the author’s Ph.D. thesis at the University of Victoria, BC, Canada. E-mail address:ibarra@cs.depaul.edu.

0166-218X/$ – see front matter©2009 Elsevier B.V. All rights reserved.

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The clique-separator graph is also different from the clique laminar tree for ptolemaic graphs [26], which are the intersection of chordal graphs and distance hereditary graphs. The clique laminar tree is a directed tree where the nodes are the sets of nonempty intersections of the maximal cliques and the arcs represent the containment relations between the sets. In a ptolemaic graph, the sets have the property that for any two sets, either they are disjoint or one is contained in the other, which ensures that the clique laminar tree is a tree. In a chordal graph, neither the maximal cliques nor the minimal vertex separators have this property.

The structural properties ofGwhenGis an interval graph have been used to develop the first dynamic graph algorithm to recognize interval graphs [15]. Adynamic graph algorithmmaintains a solution to a graph problem as the graph undergoes a series of small changes, such as single edge insertions or edge deletions. For every change, the algorithm updates the solution faster than recomputing the solution from scratch, i.e., with no previously computed information. The algorithm in [15] maintains the clique-separator graph and supports updates (edge insertions and edge deletions) that produce an interval graph and queries that ask whether a particular update is supported. The algorithm runs inO

(

nlogn

)

time per update or query, wherenis the number of vertices in the graph. The structural properties ofGwhenGis a proper interval graph have been used to simplify the algorithm in [15] so that it recognizes proper interval graphs inO

(

logn

)

time per update or query [16]. This algorithm is simpler than the algorithm in [13] and it matches its running time for edge updates and connectivity queries. In comparison, algorithms that recognize an interval graph or proper interval graph from scratch run inO

(

m

+

n

)

time, wheremis the number of edges andnis the number of vertices in the graph [3,13]. (All the running times in this paper are worst-case.)

We present the dynamic graph algorithm for interval graphs in a different paper [15] because it requires the introduction of a data structure (thetrain tree) that makes its length substantial. Also, this paper discusses classes of graphs that are unrelated to interval graphs.

In Section2, we review graph terminology and properties of chordal graphs. In Section3, we present properties of the clique-separator graph of a chordal graph. In Sections4–6, we present properties of the clique-separator graph of an interval graph, proper interval graph, and split graph, respectively. In Section7, we present an algorithm that constructs the clique-separator graph of a chordal graph inO

(

n3

)

time and of an interval graph inO

(

n2

)

time. In Section8, we consider other subclasses of chordal graphs.

2. Preliminaries

Throughout this paper, letG

=

(

V

,

E

)

=

(

V

(

G

),

E

(

G

))

be a simple, undirected graph and letn

= |

V

|

andm

= |

E

|

. For a setS

V, the subgraph ofG inducedbySisG

[

S

] =

(

S

,

E

(

S

))

, whereE

(

S

)

= {{

u

, v

} ∈

E

|

u

, v

S

}

. For a setS

V,G

S denotesG

[

V

S

]

. AcliqueofGis a set of pairwise adjacent vertices ofG. Amaximal cliqueofGis a clique ofGthat is not properly contained in any clique ofG.Fig. 1(a) shows a graph with maximal cliques

{

x

,

y

,

z

}

,

{

y

,

z

, w

}

,

{

u

,

z

}

, and

{

v,

z

}

.

LetS

V.Sis aseparatorofGif there exist two vertices that are connected inGand not connected inG

S.Sis aminimal separatorofGifSis a separator ofGthat does not properly contain any separator ofG. Foru

, v

V,Sis au

v

-separatorofGif u

, v

are connected inGand not connected inG

S.Sis aminimal vertex separatorofGif for someu

, v

V,Sis au

v

-separator ofGthat does not properly contain anyu

v

-separator ofG. Every minimal separator is a minimal vertex separator, but not vice versa.Fig. 1(a) shows a graph with minimal separator

{

z

}

and minimal vertex separators

{

z

}

and

{

y

,

z

}

. Foru

, v

V,S separates u

, v

ifSis au

v

-separator ofG. ForX

,

Y

V,S separates X

,

YifSis au

v

-separator ofGfor everyu

Xand

v

Y. A graph ischordal(ortriangulated) if every cycle of length greater than 3 has achord, which is an edge joining two nonconsecutive vertices of the cycle. A graphGis chordal if and only ifGhas aclique tree, which is a treeTon the maximal cliques ofGwith theclique intersection property: for any two maximal cliquesK andK0, the setK

K0is contained in every maximal clique on theKK0path inT[5,11,27]. The clique tree is not necessarily unique. Blair and Peyton [2] discuss

various properties of clique trees, including the following correspondence between the edges ofT and the minimal vertex separators ofG. (In [2],Gis a connected chordal graph and then the clique intersection property implies thatK

K0

6= ∅

for every

{

K

,

K0

} ∈

E

(

T

)

.)

Theorem 1 ([14,19]).Let G be a chordal graph with clique tree T . Let S

6= ∅

be a set of vertices of G. (1) S is a minimal vertex separator of G if and only if S

=

K

K0for some

{

K

,

K0

} ∈

E

(

T

)

.

(2) If S

=

K

K0for some

{

K

,

K0

} ∈

E

(

T

)

, then S separates K

S and K0

S and moreover, S is a minimal u

v

-separator for any u

K

S and

v

K0

S.

It follows that for any clique treeT of a chordal graphG, the nodes and edges ofT correspond to the maximal cliques and minimal vertex separators ofG, respectively. A maximal clique corresponds to exactly one node ofT and a minimal vertex separator may correspond to more than one edge ofT. Furthermore, since a chordal graphGhas at mostnmaximal cliques [9],Ghas at mostn

1 minimal vertex separators.

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Fig. 1. A chordal graphGand its clique-separator graphG.

Interval graphs, proper interval graphs, and split graphs are subclasses of chordal graphs and will be defined in the corresponding sections. Golumbic [12] and McKee and McMorris [20] discuss these classes in the context of perfect graphs and intersection graphs, respectively. Brandstädt, Le, and Spinrad [4] also survey these classes and Johnson [17] discusses them with respect to the hardness of problems.

We will use some terms from partially ordered set theory. LetFbe a family of sets. ForS

F,Sis aminimal element of Fif noS0

FsatisfiesS0

S, andSis amaximal element of Fif noS0

FsatisfiesS

S0.Fis achainif for anyS

,

S0

F,

we haveS

S0orS0

S.Fis anantichainif for any distinctS

,

S0

F, we haveS

6⊆

S0andS0

6⊆

S.

3. The clique-separator graph of a chordal graph 3.1. The clique-separator graph

LetGbe a chordal graph. Theclique-separator graphGofGhas the following nodes, (directed) arcs, and (undirected) edges.

Ghas aclique node Kfor each maximal cliqueKofG.

Ghas aseparator node Sfor each minimal vertex separatorSofG.

Each arc is from a separator node to a separator node.Ghas arc

(

S

,

S00

)

ifS

S00and there is no separator nodeS0such

thatS

S0

S00.

Each edge is between a separator node and a clique node.Ghas edge

{

S

,

K

}

ifS

Kand there is no separator nodeS0

such thatS

S0

K.

Throughout this paper, we refer to theverticesofGand thenodesofGand we use lowercase variables for vertices and uppercase variables for nodes. We identify ‘‘maximal clique’’ and ‘‘clique node’’ and identify ‘‘minimal vertex separator’’ and ‘‘separator node’’.Fig. 1shows a chordal graphGand its clique-separator graphG, which has clique nodesK1

= {

x

,

y

,

z

}

,

K2

=

{

y

,

z

, w

}

,

K3

= {

u

,

z

}

,

K4

= {

v,

z

}

and separator nodesS1

= {

y

,

z

}

andS2

= {

z

}

.

LetSandS0be distinct separator nodes ofG. If

(

S

,

S0

)

is an arc ofG, thenSis animmediate predecessorofS0andS0is an immediate successorofS, denotedS

S0. If there is a directed path fromStoS0inG, thenSis apredecessorofS0andS0is asuccessorofS, denotedS S0.S S0if and only ifS

S0. LetKbe a clique node ofG. If

{

S

,

K

}

is an edge ofG, thenSis

aneighborofKandSisadjacenttoKand vice versa. The number of neighbors of a node is itsdegree. IfSis a neighbor of Kor there is a directed path fromSto a neighbor ofKinG, thenSis apredecessorofKandKis asuccessorofS, denoted S K.S Kif and only ifS

K. The set of immediate predecessors ofS, the set of immediate successors ofS, and the set of neighbors ofKare antichains.

Observation.For every separator nodeS, there are at least two edges, or two arcs, or an edge and an arc, from S. By

Theorem 1(1)S

=

K

K0for some maximal cliquesKandK0, and thusS KandS K0. If the observation is false, then there is exactly one arc

(

S

,

S0

)

fromS, whereS0 KandS0 K0. But thenS

S0

K

K0, a contradiction.

The following is a simple property of the clique-separator graph that follows from the clique intersection property of clique trees.

Lemma 2. Let G be a chordal graph with clique-separator graphG. For any clique node K and any node N

6=

K , if N

K

6= ∅

, then N

K is contained in some neighbor of K inG.

Proof. LetTbe a clique tree ofG. We haveN

K0for some maximal cliqueK0

6=

KofG. By the clique intersection property, N

K

K0

K

K00

Kfor some neighborK00ofKinT. SinceN

K

6= ∅

,K00

K

6= ∅

. ThenS

=

K00

Kis a separator

node ofGbyTheorem 1(1). SinceS

K,Sis contained in some neighbor ofKinG.

For a set of nodesSofG, letV

(

S

)

= {

v

V

|

v

NandN

S

}

. For a subgraphHofG, letV

(

H

)

= {

v

V

|

v

NandNis a node ofH

}

.Hisconnectedif the underlying undirected graph ofH(obtained by replacing every arc with an edge) is connected.Gis connected if and only ifGis connected. If a subgraphHofGis connected, thenG

[

V

(

H

)

]

is connected, but the converse is not true in general.

AboxofGis a connected component of the undirected graph obtained by deleting the arcs ofG. ByLemma 2, a box of Gcontains exactly one clique nodeKand no separator nodes if and only ifKis a complete connected component ofG. We

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Fig. 2. A chordal graphG.

Fig. 3. The clique-separator graphGof the chordal graph inFig. 2.

will show that every box is a tree and so we extend the definition of the clique intersection property to a treeTon a set of nodes in the natural way: for any two nodesNandN0ofT, the setN

N0is contained in every node on theNN0path inT.

Fig. 2shows a chordal graphGandFig. 3shows its clique-separator graphG, where separator nodesS0

,

S1

,

S2have size 1, S8has size 3, and every otherSihas size 2. The boxes ofGareB1

,

B2

,

B3

,

B4

,

B5, andB6.

The graphGcis obtained fromGby contracting each box into a single node and replacing multiple arcs by a single arc. We will denote a box ofGand the corresponding node inGcby the same variable. We will show thatGcis a directed acyclic graph, which meansGchas a topological sort

σ

. Given

σ

, we will assume the boxes ofGare indexed so that

σ

=

(

B1

,

B2

, . . . ,

Bb(G)

)

, where b

(

G

)

is the number of boxes ofG. Let G

(σ,

i

)

be the subgraph of Ginduced by the nodes inBi

,

Bi+1

, . . . ,

Bb(G). We will writeG

(σ ,

i

)

asG

(

i

)

if the topological sort

σ

is clear and we will refer to

σ

as a topological sort ofG. InFig. 3,

σ

=

(

B1

,

B2

,

B3

,

B4

,

B5

,

B6

)

is a topological sort ofG.

For a separator nodeS ofG, letPreds

(

S

)

= {

S0

|

S0

=

SorS0 S

}

. We sayS dividesnodesNandN0ifNandN0 are contained in the same connected component ofGand in different connected components ofG

Preds

(

S

)

. InFig. 3,S4 dividesK5andK7andS5dividesK8andK9. We use ‘‘separates’’ when referring toGand ‘‘divides’’ when referring toG. 3.2. The main theorem

Theorem 3. Let G be a chordal graph with clique-separator graphGand letGhave topological sort

σ

. Let S be a separator node of G.

(1) G

S has connected components G1

,

G2

, . . . ,

GkandG

Preds

(

S

)

has connected componentsG1

,

G2

, . . . ,

Gksuch that k

>

1

and for every1

i

k, V

(

Gi

)

=

V

(

Gi

)

S.

(2) Every box is a tree with the clique intersection property and the set of its separator nodes is an antichain.Gcis a directed acyclic graph.

(3) For every1

i

<

j

b

(

G

)

, if S divides nodes N and N0inG

(

j

)

, then S divides N and N0inG

(

i

)

.

(4) For every1

i

<

j

b

(

G

)

and every connected componentHof G

(

j

)

, at most one separator node of box Biis a predecessor

of any node inH.

(5) For every1

j

b

(

G

)

and every connected component Hof G

(

j

)

, G

[

V

(

H

)

]

is a connected chordal graph with clique-separator graphH.

Proof. Since the proof applies to each connected component ofG, we assumeGis connected. We constructGinductively. If Gis a complete graph, thenGis a single clique node and the theorem is trivially true. Otherwise, letSbe a minimal separator ofG, which is also a minimal vertex separator ofG. ThenSis a clique [2,7,12]. LetV1

,

V2

, . . . ,

Vkbe the vertex sets of the connected components ofG

S,k

>

1. LetG1

,

G2

, . . . ,

Gkbe the subgraphs ofGinduced byV1

S

,

V2

S

, . . . ,

Vk

S, respectively (Fig. 4).

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Fig. 4. Chordal graphG.

Fig. 5. Clique-separator graphG.

Since each ofG1

,

G2

, . . . ,

Gkis connected and chordal (because an induced subgraph of a chordal graph is chordal), by induction the theorem holds for each of the corresponding clique-separator graphsG1

,

G2

, . . . ,

Gk. By the choice ofS, a separator of anyGiis a separator ofGand a minimal vertex separator of anyGiis a minimal vertex separator ofG.

SupposeSis a maximal clique of someGi. ThenSis adjacent to some maximal cliqueKin a clique tree ofGiandS

K

S. SinceGiis connected,S

K

6= ∅

. Then byTheorem 1(1),S

Kis a minimal vertex separator ofGiand thus a minimal vertex separator ofG, contradicting the minimality ofS. Therefore,Sis not a maximal clique of anyGi, or equivalently,Sis properly contained in at least one maximal clique of eachGi.

SinceSis neither a maximal clique nor a separator of anyGi,Sis not a node of anyGi. Then theGis are node-disjoint. ConsiderGiandGjfor anyi

6=

j. Since a minimal vertex separator ofGicontains at least one vertex ofVi,Gdoes not contain an edge

{

S0

,

K0

}

withS0andK0respectively inG

iandGj. Similarly,Gdoes not contain an arc

(

S0

,

S00

)

withS0andS00respectively inGiandGj. Therefore,Ghas no edge or arc between a node ofGiand a node ofGj. It follows that we may constructGfrom theGis by creating separator nodeSand adding edges and arcs fromSto nodes in theGis. We do this in two steps, as follows. Edge Step: For eachisuch thatSis contained in exactly one maximal cliqueKofGi, add an edge betweenSandKand add no arcs fromS. (IfSis contained in a minimal vertex separator ofGi, thenSis contained in at least two maximal cliques ofGi.) Arc Step: For eachisuch thatSis contained in maximal cliquesK1

,

K2

, . . . ,

KlofGi

,

l

>

1, letTibe a clique tree ofGi. LetMibe the set of minimal vertex separators ofGicorresponding to edges on a path inTibetween any two ofK1

,

K2

, . . . ,

Kl. By the clique intersection property,Sis contained in each element ofMi. Add an arc fromSto each minimal element ofMi and add no edges fromS. This yields the clique-separator graphGofG.

Without loss of generality, assume theGis are indexed so that theGis in the Edge Step areG1

, . . . ,

Gjand theGis in the Arc Step areGj+1

, . . . ,

Gk, where 0

j

k(Fig. 5). Thus, if 1

i

j, we added exactly one edge fromSto a clique node of

Gi, and ifj

+

1

i

k, we added one or more arcs fromSto certain separator nodes ofGi. We now prove each part of the theorem.

1. We defined eachGito be the subgraph ofGinduced byVi

S, whereas part 1 of the theorem defines eachGito be the subgraph ofGinduced byVi. SinceS

=

Preds

(

S

)

,Ssatisfies part 1. Now letS0be a separator node of someGi. By induction,

S0satisfies part 1 inGi. We must show thatS0satisfies part 1 inG.

Suppose 1

i

j, so thatGhas exactly one edge fromSto a clique node ofGi. ThenS

6∈

Preds

(

S0

)

, which implies

Gi

Preds

(

S0

)

andG

Preds

(

S0

)

have the same number of connected components. NowSis a minimal separator (implying every vertex ofSis adjacent to at least one vertex of everyVi) andSis a clique. SinceS

6⊂

S0,Gi

S0andG

S0have the same number of connected components. It follows thatS0satisfies part 1 inG. Supposej

+

1

i

k, so thatGhas arcs fromSto

separator nodes ofGi. IfS

6∈

Preds

(

S0

)

, thenS

6⊂

S0and the same argument shows thatS0satisfies part 1 inG. IfS

Preds

(

S0

)

,

thenS

S0and a similar argument shows thatS0satisfies part 1 inG.

2. In the Edge Step, adding an edge betweenSand one box of each ofG1

, . . . ,

Gjcauses these boxes to merge into a boxB containingS; since each box is a tree,Bis a tree. (Ifj

=

0, thenBcontains onlyS.) SinceSis contained in each of its neighbors inB, it follows thatBhas the clique intersection property. By the choice ofS,Bdoes not contain separator nodesS0andS00

such thatS0

S00. Thus, the set of separator nodes ofBis an antichain. Since every arc added in the Arc Step is an arc from S, it follows thatGcis a directed acyclic graph.

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Fig. 6. A chordal graphGand its clique-separator graphG.

3. Let

σ

=

(

B1

,

B2

, . . . ,

Bb(G)

)

. SinceGis connected and not complete, each boxBihas at least one separator node by

Lemma 2. A separator node ofB1has no predecessors and thus it is a minimal separator ofG. Therefore, we may assumeS is a separator node ofB1without affecting the proof of parts 1 and 2. Then each ofB2

, . . . ,

Bb(G)is a box of someGi.

Let

σ

ibe the restriction of

σ

toGi

,

1

i

k. In other words,

σ

iis the subsequence of

σ

that contains the boxes ofGi; if 1

i

j, then the box ofGicontained inB1is namedB1in

σ

i. Then

σ

iis a topological sort ofGi. Observe that forl

>

1,

His a connected component ofG

(σ,

l

)

if and only ifHis a connected component ofGi

i

,

l

)

for somei. We will use this observation several times in the rest of the proof.

We now show that for every 1

i0

<

j0

b

(

G

)

, if a separator nodeS0divides nodesNandN0inG

(σ,

j0

)

, thenS0divides NandN0inG

(σ ,

i0

)

, proving part 3. SupposeS0dividesNandN0inG

(σ,

j0

)

. ThenNandN0are contained in a connected

component ofG

(σ ,

j0

)

and by the observation, in a connected component ofG

i

,

j0

)

for somei. It follows thatS0divides

N andN0inGi

i

,

j0

)

. By induction,S0dividesN andN0inGi

i

,

i0

)

. Ifi0

>

1, then by the observation, every connected component ofGi

i

,

i0

)

is a connected component ofG

(σ ,

i0

)

and thusS0dividesNandN0inG

(σ ,

i0

)

. Supposei0

=

1. ThenS0 dividesNandN0inG

i

i

,

1

)

=

Giand we must showS0dividesNandN0inG

(σ ,

1

)

=

G.

If 1

i

j, then the definition of the Edge Step implies thatS0dividesNandN0inG. Ifj

+

1

i

k, then supposeG

has arcs fromSto separator nodesS1andS2in different connected components ofGi

Preds

(

S0

)

. (IfGhas no such arcs, then

S0dividesNandN0inG.) ThenS

S

1andS

S2. Now if there is a vertex

v

S

S0, then

v

S1

S0and

v

S2

S0. ButS0dividesS

1andS2inGiand by part 1,S0separatesS1

S0andS2

S0inGi, a contradiction. Therefore,S

S0. Then the definition of the Arc Step implies thatS S0inGand thusS

Preds

(

S0

)

. Then every connected component ofGi

Preds

(

S0

)

is a connected component ofG

Preds

(

S0

)

, which impliesS0dividesNandN0inG.

4. LetHbe a connected component ofG

(σ,

j0

),

1

<

j0

b

(

G

)

. By the observation,His a connected component ofG

i

i

,

j0

)

for somei. Consider the boxes ofG. For every boxBi0ofG, 1

<

i0

<

j0, ifB

i0is a box ofGi, then by induction,Bi0 contains at most one predecessor of a node inH, and otherwise,Bi0contains no predecessor of a node inH. Lastly,B1contains at most one predecessor of a node inHby induction (if 1

i

j) or by the definition of the Arc Step (ifj

+

1

i

k).

5. LetHbe a connected component ofG

(σ ,

j0

),

1

j0

b

(

G

)

. Ifj0

=

1, thenH

=

GandG

[

V

(

H

)

] =

Gand we are done.

Supposej0

>

1. By the observation,His a connected component ofG

i

i

,

j0

)

for somei. By induction,Gi

[

V

(

H

)

]

is a connected chordal graph with clique-separator graphH. SinceV

(

H

)

V

(

Gi

)

=

V

(

Gi

)

, it follows thatG

[

V

(

H

)

] =

Gi

[

V

(

H

)

]

. Hence,

G

[

V

(

H

)

]

is a connected chordal graph with clique-separator graphH.

The Main Theorem 5 is not used in the rest of this paper, but it is necessary for [15]. Notice thatG

(

j

)

is not always the clique-separator graph of the chordal graphG

[

V

(

G

(

j

))

]

. InFig. 6, for example,G

(

2

)

is not the clique-separator graph of G

[

V

(

G

(

2

))

] =

G.

We extend the definition of divides to sets of nodesSandS0ofGas follows:S dividesSandS0ifS

Preds

(

S

)

and S0

Preds

(

S

)

are contained in the same connected component ofGand in different connected components ofG

Preds

(

S

)

. We extend the definition of divides to subgraphsHandH0ofGin the same way. We need one more definition:S strongly dividesnodesNandN0ifS NandS N0andSdividesNandN0. InFig. 3,S

4dividesK5andK7but does not strongly divideK5andK7, whereasS1strongly dividesK5andK7.

Corollary 4. Let G be a chordal graph with clique-separator graphG. Let S be a separator node of G. (1) Let N and N0be nodes of G. Then S divides N and N0if and only if S separates N

S and N0

S.

LetSandS0be sets of nodes ofG. Then S dividesSandS0if and only if S separates V

(

S

)

S and V

(

S0

)

S. LetHandH0be subgraphs of G. Then S dividesHandH0if and only if S separates V

(

H

)

S and V

(

H0

)

S. (2) If S has distinct neighbors N and N0, or S has neighbor N and successor N0, then S strongly divides N and N0.

(3) S strongly divides nodes N and N0if and only if S

=

N

N0and S separates N

S and N0

S.

(4) S strongly divides clique nodes K and K0if and only if S

=

K

K0for some edge

{

K

,

K0

}

in a clique tree of G. Proof. (1) By the Main Theorem 1.

(2) LetBibe the box containingSin some topological sort ofG. By the Main Theorem 4,SdividesNandN0inG

(

i

)

. Ifi

=

1, thenG

(

i

)

=

G, and ifi

>

1, then by the Main Theorem 3,SdividesNandN0inG. SinceS NandS N0, it follows thatSstrongly dividesNandN0inG.

(3) (

) By part 1,SseparatesN

SandN0

S. SinceS NandS N0, it follows thatS

N

N0. IfS

N

N0, then some vertex is in bothN

SandN0

S, a contradiction. Thus,S

=

N

N0. (

) By part 1,SdividesNandN0. Since S

N

N0, it follows thatS NandS N0.

(7)

(4) (

) ByTheorem 1(2),SseparatesK

SandK0

S. By part 3,Sstrongly dividesKandK0. (

) LetTbe any clique tree ofG. If

{

K

,

K0

}

is an edge ofT, we are done. Otherwise, letK

=

K

1

,

K2

, . . . ,

Kj

=

K0be theKK0path inT, wherej

3. SinceT has the clique intersection property,S

Ki and thusS Kifor every 1

i

j. ThenK1

,

K2

, . . . ,

Kjare in the same connected component ofG. NowSdividesK1andKj. ThenK1andKjare in different connected components of

G

Preds

(

S

)

and for some 1

i

j

1,KiandKi+1are in different connected components ofG

Preds

(

S

)

. ThenS dividesKiandKi+1and thusSstrongly dividesKiandKi+1. By part 3,S

=

Ki

Ki+1

=

K1

Kj. Then the tree obtained fromTby deleting edge

{

Ki

,

Ki+1

}

and adding edge

{

K1

,

Kj

}

is a clique tree ofG. (A tree is a clique tree ofGif and only if it is a maximum spanning tree of theweighted clique intersection graphin [1,2]. In this graph, the nodes are the maximal cliques ofGand the graph has edge

{

K

,

K0

}

with weight

|

K

K0

|

if and only ifK

K0

6= ∅

.)

Remark.Corollary 4(4) implies that for any connected chordal graphGwith clique nodesKandK0,Ghas a clique tree with

edge

{

K

,

K0

}

if and only ifGhas a separator node that strongly dividesKandK0. (We needGto be connected because ifK andK0are in different connected components ofG, thenGhas a clique tree with edge

{

K

,

K0

}

but no separator node equals K

K0

= ∅

.)

4. The clique-separator graph of an interval graph

Aninterval graphis a graph where each vertex can be assigned an interval on the real line so that two vertices are adjacent if and only if their assigned intervals intersect; such an assignment is aninterval representation. A graph is an interval graph if and only if it has aclique path, which is a clique tree that is a path [9]. Interval graphs are a proper subclass of chordal graphs. Since interval graphs model many problems involving linear arrangements, interval graphs have as many applications as chordal graphs. Application areas include archeology, computational biology, file organization, partially ordered sets, and psychology [4,12,20].

In this section, we present properties of the clique-separator graphGwhenGis an interval graph. A pathPon a set of nodes ofG(not necessarily a subgraph ofG) is ac.i.p.path ifPhas the clique intersection property. We will use the following characterization, which is straightforward to prove from the clique path characterization. LetGbe a chordal graph with clique-separator graphG. ThenGis an interval graph if and only if there is ac.i.p.path on the nodes ofG.

For a clique treeTofG, letST

(

K

)

= {

K

K0

| {

K

,

K0

} ∈

E

(

T

)

andK

K0

6= ∅}

, which is a set of minimal vertex separators byTheorem 1(1). In the following lemma, part 4 is a property of clique trees that follows from part 1.

Lemma 5. Let G be an interval graph with clique-separator graphG.

(1) For every node, the set of its predecessors can be partitioned into at most two chains.

(2) Every clique node has at most two neighbors. Every separator node has at most two immediate predecessors. (3) There are O

(

n

)

nodes, edges and arcs inG.

(4) For every maximal clique K and clique tree T of G,ST

(

K

)

can be partitioned into at most two chains.

Proof. (1) By the characterization, there is ac.i.p.path that contains every node ofG. LetNbe any node and letSbe the set of its predecessors. IfScontains an antichain

{

S1

,

S2

,

S3

}

, then sinceS1

N,S2

N,S3

N, there is noc.i.p.path containingS1

,

S2

,

S3, andN, a contradiction. Thus,

(

S

,

)

is a partially ordered set whose largest antichain has size at most 2. By Dilworth’s Theorem, which states that the size of the largest antichain is equal to the size of the smallest chain decomposition,Scan be partitioned into at most two chains.

(2) By part 1.

(3) SinceGis a chordal graph with at mostnmaximal cliques andn

1 minimal vertex separators,Ghas at most 2n

1 nodes. By the Main Theorem 2,Ghas at most 2n

2 edges. By part 2,Ghas at most 2n

2 arcs.

(4) By part 1 because the elements ofST

(

K

)

are predecessors ofKinG.

A pathPon a set of nodes ofG(not necessarily a subgraph ofG) isconstrainingifPis the uniquec.i.p.path on the set of nodes inP. For pathsPandQ on sets of nodes ofG,Pis asubsequenceofQ if, when the paths are viewed as sequences,P or its reverse is a subsequence ofQ. Then a constraining pathPis a subsequence of everyc.i.p.path that contains the nodes ofP.

Lemma 6. Let G be an interval graph with clique-separator graphG. Let B be a box of G. (1) Every path in B between two separator nodes is a constraining path.

(2) Every separator node of B has at most two neighbors that are internal nodes of B.

Proof. By the Main Theorem 2, every path contained inBis ac.i.p.path and the set of separator nodes inBis an antichain. Observe that ifBcontains a separator nodeSand a clique nodeKsuch thatS

K, thenSis a neighbor ofK, or elseS

S0

for some neighborS0ofK, a contradiction.

(1) Let

(

S1

,

K1

,

S2

,

K2

,

S3

)

be a path contained inB. Then

{

S1

,

S2

,

S3

}

is an antichain. SinceS1

K1andS2

K1, it follows that

(

S1

,

K1

,

S2

)

is a constraining path. By the observation,S1

6⊂

K2. Then

(

S1

,

K1

,

S2

,

K2

)

and

(

S1

,

K1

,

K2

,

S2

)

are the onlyc.i.p. paths on these four nodes. SinceS3

K2, it follows that

(

S1

,

K1

,

S2

,

K2

,

S3

)

is a constraining path. Applying induction with a similar argument shows that every path inBbetween two separator nodes is a constraining path.

(8)

Fig. 7. A separator nodeSwith three neighbors that are internal nodes.

Fig. 8. Proof ofTheorem 8.

(2) Let S be a separator node with three neighbors that are internal nodes of box B (Fig. 7). Then B contains paths

(

S

,

K1

,

S1

), (

S

,

K2

,

S2

), (

S

,

K3

,

S3

)

and

{

S

,

S1

,

S2

,

S3

}

is an antichain. By part 1,

(

S1

,

K1

,

S

,

K2

,

S2

)

is a constraining path and by the observation,S1

6⊂

K3andS2

6⊂

K3. It follows thatP

=

(

S1

,

K1

,

K3

,

K2

,

S2

)

is a constraining path. SinceS3

K3, there is noc.i.p.path containingS3and the nodes inP. But by the characterization, there is ac.i.p.path containingS3and the nodes inP, a contradiction.

Aleaf clique node is a clique node that has degree 1, or equivalently, a clique node that is a leaf of some box. Thebody of a boxB, denotedbody

(

B

)

, is the subgraph ofBobtained by deleting the leaf clique nodes ofB.Lemmas 5and6yield the following theorem, which summarizes the properties ofGwhenGis an interval graph.

Theorem 7. Let G be an interval graph with clique-separator graphG. Then for every box B, body

(

B

)

is a constraining path whose endpoints are separator nodes. Moreover, every separator node has at most two immediate predecessors and there are O

(

n

)

nodes, edges and arcs inG.

Theorem 7is used by the dynamic graph algorithm for interval graphs in [15] to decide whetherGis an interval graph. The algorithm requires additional results since the condition inTheorem 7is necessary but not sufficient forGto be an interval graph.

5. The clique-separator graph of a proper interval graph

Aproper interval graph(orunit interval graph) is an interval graph with an interval representation where no interval is properly contained in another. Proper interval graphs are a proper subclass of interval graphs. For example,K1,3(the graph consisting of one vertex adjacent to three nonadjacent vertices) is an interval graph but not a proper interval graph. In fact, an interval graph is a proper interval graph if and only if it has no induced subgraph isomorphic toK1,3[23]. Anindependent setofGis a set of pairwise nonadjacent vertices.

The next theorem presents the properties ofGwhenGis a proper interval graph. IfGis not connected, the theorem can be applied to each of its connected components.

Theorem 8. Let G be a connected chordal graph with clique-separator graphG. Then G is a proper interval graph if and only if Ghas exactly one box and this box is a path P such that there do not exist vertices u

K1

K3and

v

K2

(

K1

K3

)

where

(

K1

,

K2

,

K3

)

is a subsequence of P.

Proof. (

) SupposeGhas an arc

(

S

,

S0

)

. ByTheorem 1(1) andCorollary 4(4), there are clique nodesK

1andK2such that S0strongly dividesK1andK2inG. ThenS0 K1andS0 K2and thusK1 andK2are in one connected component of G

Preds

(

S

)

(Fig. 8). Then byCorollary 4(4),S K3for some clique nodeK3 such that

{

K1

,

K2

}

andK3are in different connected components ofG

Preds

(

S

)

. ThenSdivides

{

K1

,

K2

}

andK3. Lety1

K1

S0

,

y2

K2

S0

,

y3

K3

S. By

Corollary 4(1),S0separatesy1andy2andSseparates

{

y1

,

y2

}

andy3. Then

{

y1

,

y2

,

y3

}

is an independent set ofG. Now any x

S

S0is adjacent to each ofy

1

,

y2

,

y3, which meansGhas an inducedK1,3, a contradiction. Thus,Ghas no arc. SinceG is connected,Ghas exactly one box.

SupposeSis a separator node with three neighborsK1

,

K2

,

K3. Letyi

Ki

S

,

i

=

1

,

2

,

3. ByCorollary 4(1) and4(2),S pairwise separatesy1

,

y2

,

y3. Then

{

y1

,

y2

,

y3

}

is an independent set ofG. Since anyx

Sis adjacent to each ofy1

,

y2

,

y3,G has an inducedK1,3, a contradiction. Thus, every separator node has at most two neighbors. Since every clique node has at most two neighbors (Lemma 5(2)), the box ofGis a pathP.

(9)

Fig. 9. Proof ofTheorem 8.

Fig. 10. Proof ofLemma 9.

Suppose there exist verticesu

K1

K3and

v

K2

(

K1

K3

)

where

(

K1

,

K2

,

K3

)

is a subsequence ofP. LetS1andS3 be the neighbors ofK1andK3that are on theK1–K3path inP, respectively (Fig. 9). Letx

K1

S1andy

K3

S3. Since

v

6∈

K1

K3, it follows that

v

6∈

S1

S3. ByCorollary 4(1),S1separatesxand

v

,S3separates

v

andy, andS1(orS3) separates xandy. Then

{

x

, v,

y

}

is an independent set ofG. Sinceu

K1

K3andPhas the clique intersection property,uis adjacent to each ofx

, v,

y. ThenGhas an inducedK1,3, a contradiction.

(

) SinceGhas exactly one box and this box is a pathP, deleting the separator nodes fromPyields a clique path of G. ThenGis an interval graph. SupposeGis not a proper interval graph. ThenGcontains verticesu

, v

1

, v

2

, v

3such thatu adjacent to each of

v

1

, v

2

, v

3and

{

v

1

, v

2

, v

3

}

is an independent set ofG. Then

v

1

K1

, v

2

K2

, v

3

K3for distinct clique nodesK1

,

K2

,

K3ofP. Without loss of generality, assume that

(

K1

,

K2

,

K3

)

is a subsequence ofPand thatK1andK3are as close as possible onP. LetS1andS3be the neighbors ofK1andK3on theK1–K3path inP, respectively (Fig. 9). ByCorollary 4(1), S1separatesK1

S1andK2

S1. Since

v

1

K1

S1and

v

2

K2

S1anduis adjacent to

v

1and

v

2, it follows thatu

S1. The symmetric argument shows thatu

S3, which meansu

K1

K3. Now

v

2is adjacent to neither

v

1nor

v

3and thus

v

2

6∈

K1

K3. Then

v

2

K2

(

K1

K3

)

, a contradiction. Hence,Gis a proper interval graph.

Theorem 8characterizes proper interval graphs in terms of the clique-separator graph; its corollary characterizes proper interval graphs in terms of the clique tree. The corollary requires the following lemma, which referencesTheorem 1(1). Lemma 9. Let G be a connected chordal graph with clique tree T . Then T is the unique clique tree of G if and only if the family of minimal vertex separators corresponding to the edges of T is an antichain.

Proof. We prove the contrapositive of both directions. SinceGis connected, the clique intersection property implies that K

K0

6= ∅

for every edge

{

K

,

K0

} ∈

E

(

T

)

.

(

) SupposeGhas distinct clique treesT andT0. Then there are maximal cliquesKandK0that are adjacent inT0and not adjacent inT. ThenS

=

K

K0is a minimal vertex separator. Now theKK0path inT contains (at least) two edges,

which correspond to minimal vertex separatorsS0andS00. SinceT has the clique intersection property,S

S0andS

S00.

We claim that the family of minimal vertex separators corresponding to the edges ofTis not an antichain. IfS0

=

S00, then

the claim holds, and ifS0

6=

S00, thenS

S0orS

S00, and again the claim holds.

(

) Suppose the family of minimal vertex separators corresponding to the edges ofTis not an antichain. ThenTcontains distinct edges

{

K1

,

K2

}

and

{

K3

,

K4

}

such thatS

=

K1

K2

K3

K4

=

S0. Without loss of generality, assume that

(

K1

,

K2

,

K3

,

K4

)

is a subsequence of a path contained inT; we may haveK2

=

K3(Fig. 10). LetT0be the tree obtained fromPby deleting edge

{

K1

,

K2

}

and adding edge

{

K1

,

K4

}

. We claim thatT0has the clique intersection property. To show this, letKandK0be maximal cliques such that theKK0path inT0contains edge

{

K

1

,

K4

}

. Then theKK0path inTcontains edge

{

K1

,

K2

}

, which implies thatK

K0

S

S0. It follows that the claim holds and thusTandT0are distinct clique trees

ofG.

Corollary 10. Let G be a connected chordal graph. Then G is a proper interval graph if and only if G has a unique clique tree and this tree is a path P such that there do not exist vertices u

K1

K3and

v

K2

(

K1

K3

)

where

(

K1

,

K2

,

K3

)

is a subsequence of P.

Proof. (

) LetPbe the box ofGsatisfying the conditions ofTheorem 8. Then deleting the separator nodes ofPyields a clique pathP0ofG, which is a clique tree ofG. By the Main Theorem 2, the family of minimal vertex separators corresponding to the edges ofP0is an antichain. ByLemma 9,Pis the unique clique tree ofG. The rest of the statement holds because it is

the same inTheorem 8and in this corollary.

(

) ByLemma 9, the family of minimal vertex separators corresponding to the edges ofPis an antichain. LetP0be the path obtained fromPby inserting the minimal vertex separatorK

K0between every consecutive pairKandK0of maximal

cliques inP. Since the inserted sets form an antichain, eachK

K0is not properly contained in any minimal vertex separator

or any maximal cliques other thanKandK0. Therefore,P0is a box ofG. Moreover, sinceP0contains every node ofG,P0is the only box ofG. ByTheorem 8,Gis a proper interval graph.

6. The clique-separator graph of a split graph

Asplit graphis a graph whose vertex set can be partitioned into a cliqueKand an independent setI, where either set may be empty and there may be edges between vertices inKand vertices inI. A graphGis a split graph if and only ifGand its complementG

¯

are chordal [8]. (The complement ofG

=

(

V

,

E

)

isG

¯

=

(

V

,

E

¯

)

whereE

¯

= {{

u

, v

} |

u

, v

Vandu

6=

(10)

(a) Split graphG. (b) Clique-separator graph ofG.

Fig. 11. Example ofTheorem 11(1).

(a) Split graphG. (b) Clique-separator graph ofG.

Fig. 12. Example ofTheorem 11(2).

v

and

{

u

, v

} 6∈

E

}

.) Split graphs are a proper subclass of chordal graphs and are not comparable to interval graphs or proper interval graphs.

The next theorem presents the properties ofGwhenGis a split graph.Fig. 11illustrates Property 1 of the theorem with clique nodeK

ˆ

=

K andFig. 12illustrates Property 2 of the theorem with separator nodeS

ˆ

=

K. In both figures, each Ki

=

N

(v

i

)

∪ {

v

i

}

and eachSi

=

N

(v

i

)

, whereN

(v)

is the set of neighbors of vertex

v

. A vertex isisolatedif it has degree zero. Theorem 11. Let G be a chordal graph with clique-separator graphG. Then G is a split graph if and only if Ghas either of the following properties.

(1) There is a clique nodeK such that every separator node is a predecessor of

ˆ

K and every other clique node K is a leaf that

ˆ

contains exactly one more vertex than its neighbor, unless K consists of one isolated vertex.

(2) There is a separator nodeS such that every other separator node is a predecessor of

ˆ

S and every clique node K is a leaf that

ˆ

contains exactly one more vertex than its neighbor, unless K consists of one isolated vertex.

Proof. (

) SupposeGhas Property 1. IfGhas exactly one clique node, thenGis a complete graph and we are done. Otherwise, if every clique node ofGcontains exactly one vertex, thenGis an independent set and we are done. Otherwise, letK1

, . . . ,

Kl be the clique nodes that contain more than one vertex and letS1

, . . . ,

Sl be their neighbors, respectively. EachKicontains exactly one vertex

v

inot contained inK

ˆ

. Since eachKi is a leaf,SidividesKiandKjfor anyi

6=

j. Then byCorollary 4(1),

{

v

i

|

1

i

l

}

is an independent set. Therefore, the vertex set ofGis partitioned into cliqueK

ˆ

and independent set

{

v

i

|

1

i

l

}∪

{isolated vertices}. SupposeGhas Property 2. Then a similar argument shows that the vertex set ofGis partitioned into cliqueSand independent set

{

v

i

|

1

i

l

} ∪

{isolated vertices}.

(

) Suppose the vertex set ofGcan be partitioned into cliqueKand independent setI. Observe that for every

v

I, N

(v)

∪ {

v

}

is a maximal clique, and thatKis not a maximal clique if and only if some vertex inIis adjacent to every vertex inK.

Case 1:Kis a maximal clique. Consider any

v

I. IfN

(v)

= ∅

, then

v

is an isolated vertex. Otherwise,N

(v)

KandN

(v)

is a minimalu

v

-separator for everyu

K

N

(v)

, which meansN

(v)

is a minimal vertex separator. Then inG, separator nodeN

(v)

and clique nodeN

(v)

∪{

v

}

are adjacent andN

(v)

is a predecessor of clique nodeK. It follows thatGhas Property 1 withK

ˆ

=

K. Case 2:Kis not a maximal clique. Let

v

1

, . . . , v

jbe the vertices ofIthat are adjacent to every vertex ofK,j

1, and let

v

j+1

, . . . , v

kbe the remaining vertices ofI. Ifj

=

1, then for eachj

+

1

i

k,N

(v

i

)

Kand, as before, separator

Figure

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References