Properties of Chordal Graphs

Undergraduate Research Opportunities Programme in Science

(UROPS)

Semester 2, 2001/2002

Department of Mathematics

National University of Singapore

Supervisor: A/P Tay Tiong Seng

Done by: Jaron Pow Tien Min (U002626M02)

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Acknowledgements

Very special thanks to A/P Tay Tiong Seng for agreeing in the first place to

undertake this Urops project on Chordal Graphs. Without him, I would not have been

able to dwell as deeply as I did into the topic of chordal graphs.

Taking up Graph Theory in UROPS was of my intial highest priority as it is my

favorite topic of all in tertiary mathematics, mostly also due to the fact that A/P Tay was

the lecturer for MA3233, which was the deciding factor for me liking the topic all the

way from the start, with further readings out of the syllabus and invitations to a talk by

the dean of Mathematics from Hong Kong on Graph Theory helping every bit.

I would like to thank A/P Tay again on his patience in guiding me through the

finer aspects of proving theorems and lemmas, be it in the very basics of Graph theory or

in the later dwellings on the LexBFS algorithm.

The last thanks goes to my peers in the Special Programme who have helped me

look through parts of the report and discussing with me also the many areas in Graph

Theory.

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Abstract

Graph theory started in the early 1700s where Euler discussed the problem of

whether is it possible to cross the 7 bridges of Konigsberg exactly once. Of course, the

topic of Graph theory evolved through the years such that we now have representations

like vertices, edges and cycles (note that when Euler solved the Konigsberg problem, he

did not at all use the concepts of edges and vertices at all. All of the terminology that we

use now is a result of mathematicians going deeper into the topic and implementing the

terms that they find useful to study of graph theory).

Currently, many mathematicians and computer scientists are going into graph

theory as certain branches of its study are important in their respective fields.

What we have hoped to achieve in this paper is to go deeper into the study of a

particular aspect of graph theory, and the choice was chordal graphs as it is currently

gaining popularity in computer scientists.

Chordal graphs show many links to perfect graphs and interval graphs. In this

paper is a short proof to how all interval graphs are triangulated, but more importantly,

we touched on the topic of moplexes, which serve to generalize Dirac’s theorems

regarding triangulated graphs.

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CONTENTS

1. PRELIMINARIES 5

2. INTERVAL GRAPHS 9

3. RELATIONSHIP OF TRIANGULATED GRAPHS

TO THE PERFECT ELIMINATION SCHEME 11

4. MOPLEXES IN TRIANGULATED GRAPHS 14

5. GENERALIZATION OF DIRAC’S THEOREM

TO ANY GRAPH 18

6. REFERENCES 21

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1. PRELIMINARIES

In this paper, the notations used will be as follows.

1.1 Graphs

G = (V,E) is a finite undirected graph with vertex set V and edge set E, |V| = n,

|E| = m. N(x) denotes the neighborhood of vertex x (note that it does not contain x

itself). If N(X) is the neighborhood of X where X V, N(X) = {UxєX N(x) \ X}.

1.2 Triangulated Graphs

A simple Graph G is triangulated if every cycle of length > 3 has an edge joining

2 nonadjacent vertices of the cycle. The edge is called a chord, and triangulated

graphs are also called chordal graphs.

1.3 Cliques and Simplicial Vertices

A clique of G is a set of pairwise adjacent vertices.

A vertex v of a graph G is a simplicial vertex iff the induced subgraph of N(v), is

a clique.

1.4 The Perfect Vertex Elimination Scheme

A perfect vertex elimination scheme of a graph G is an ordering {v1, v2, v3, ..., v

n } such that for 1 ≤ i ≤ n-1, vi is a simplicial vertex of the subgraph of G induced

by {vi, vi+1, vi+2, ..., v n }. It is also called a perfect scheme.

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(Remarks)

Any vertex of degree 1 is trivially simplicial.

For a tree, there exist at least 2 end vertices. Since end vertices are of degree 1

and hence trivially simplicial, every tree has at least 2 simplicial vertices. After

deleting an end vertex, we still get a tree. Therefore, every tree has a perfect

vertex elimination scheme of sequence {v1, v2, v3, ..., v n }, where vi is an end

vertex of the subgraph which is a tree induced by { vi, vi+1, vi+2 , … , vn}

1.5 Separation

A subset S V of a connected graph G is called a separator iff G(V\S) is

disconnected. The set of the connected components of G(V\S) is denoted as

CC(S). S is called an ab-separator iff a and b lie in 2 different components of

CC(S). S is called a minimal ab-separator iff S is an ab-separator and no proper

subset of S is also an ab-separator. S is called a minimal separator iff a,b є V

such that S is a minimal ab-separator.

1.6 Triangulation

A triangulated graph H = (V, E U F) is called a triangulation of G = (V, E).

The triangulation is minimal iff for any edge e of F, H’ = (V, (E U F)\{e}) is not

triangulated. F is then called a minimal fill-in.

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Unique Chord Property

A triangulation H of G is minimal iff for all e є F, e is the unique chord of some

4-cycle of H.

Chord is unique Chords are not unique

A minimal fill-in Not a minimal fill-in

Crossing edge Lemma

No edge of a minimal fill-in of G can join 2 connected components in CC(S),

where S is a clique separator of G (a clique separator is a separator that is a

clique).

Proof: If C is a clique separator of a graph G, G – C consists of at least 2

separated cut components. Take 2 vertices a and b from the 2 cut

components. Every cycle containing a and b must consists of at least 2

vertices s, t in S. Since S is a clique, the cycle is split into 2 smaller cycles

in GA and GB respectively because the cycle containing a, b is split into 2

by the s-t edge. Thus, to triangulate the graph G, the individual cycles in

GA and GB must be triangulated first. Hence, a minimal fill-in will not have

an edge that joins 2 connected components.

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Minimal separator property

Let H be a minimal triangulation of G. Any minimal ab-separator of H is also a

minimal ab-separator of G.

Proof: It can be easily deduced that any separator of H is also a separator of G.

Let S be a minimal ab-separator of H (S is a clique) and G’ be obtained from G be

inserting edges to S such that S becomes a clique. Thus, H is a triangulation of G’.

By the crossing-edge lemma, if any subset S’ of S is an ab-separator of G’, then it

is an ab-separator in H, since no edges added join 2 connected components. Thus,

S is a minimal ab-separator in G’. And since S-x is not an ab-separator in G, it is a

minimal ab-separator of G.

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2. INTERVAL GRAPHS

Definition: A graph G = (V,E) is an interval graph iff there exists an assignment to each

vertex x є V of an interval J(x) on the real number line such that x, y є E J(x) J(y)

.

Proposition

All Interval Graphs are triangulated

Proof: Assume that there exists an interval that is not triangulated. This implies that there

we can create a cycle of length greater than 3 which does not contain a chord.

There are only a few ways to construct an interval representation of a P3. Let the 3

vertices be a, b and c, with b being the vertex that is connected to both a and c.

J(a) J(c) J(b)

J(a) J(c) J(b)

J(a) J(c) J(b)

Without loss of generality, these are the only 3 interval representations of a P3.

In the latter two cases, any interval that overlaps with J(c) will also overlap with J(b).

Thus the vertex it represents will be adjacent to b. In the first case, since there is a path

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from c to a, one of the intervals representing this path must overlap with J(b) and

hencethere is a chord as well.

In order to create a true 4-cycle, an interval (or a series of them for a chordless cycle of

length greater than 4) has to be created that overlaps J(a) and J(c) but not J(b). From the

representations above, we see that it is not possible. Hence, there is no chordless cycle

that is an interval graph, which implies that all interval graphs are triangulated.

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3. RELATIONSHIP OF TRIANGULATED GRAPHS TO THE PERFECT ELIMINATION SCHEME

Theorem 3.1 If S is a minimal ab-separator, every vertex x in S must be adjacent to

some vertex a of GA and some vertex b of GB

For any x S, since S-x is not a separator, GA and GB will be connected in G-{S-x}.

Hence, there exists an a-b path which contains x. Therefore, x must be adjacent to some

vertex in GA and some vertex in GB.

Theorem 3.2 (Dirac’s Theorem) A graph is triangulated iff every minimal vertex

separator of G is a clique.

Necessity: Let the graph G be triangulated and S be a minimal separator of G. Let GA and

GB be 2 distinct components of G\S. Since S is a minimal separator, every vertex x in S

must be adjacent to some vertex of GA and some vertex of GB. Hence, for any pair x, y in

S, there exist paths P1: xa1…ary and P2: xb1…bsy where each ai є V(GA) and each

bi є V(GB). Assuming also that P1 and P2 are chosen to be of the shortest length, xa1…

arybs…b1x is a cycle of length at least 4, and so (as G is triangulated) must contain a

chord. However, as P1 and P2 are chosen to be of the shortest length, the chord must be

xy. Thus, every pair x, y in S are adjacent and S is a clique.

Sufficiency: We now have to prove that if every minimal separator of G is a clique,

every cycle of length at least 4 in G contains a chord. Assume that every minimal

separator of G is a clique. Let axby1y2… yra be a cycle C of length 4 in G. If ab were

not a chord of C, denote by S a minimal seperator that puts a and b in distinct

components of G\S.Then S must contain x and yj for some j. By hypothesis, S is a

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clique, and hence xyj is an edge of G, and therefore a chord in C. Thus,

G is triangulated.

Lemma 3.3 Every triangulated graph G has a simplicial vertex. Moreover, if G is not

complete, it has 2 nonadjacent simplicial vertices. (Dirac’s

characterization).

If G is either complete or has just 2 or 3 vertices, the lemma is trivial.

Thus, we assume that G is not complete. We shall prove the lemma by induction.

Assume that the lemma is true for all graphs with fewer vertices than G. Let S be a

minimal ab-separator, and let GA and GB be components of G\S containing a and b,

respectively, and with vertex sets A and B respectively. By the induction hypothesis, if

G[A U S] is not complete, it has 2 nonadjacent simplicial vertices. This way, since G[S]

is complete, at least one of the 2 simplicial vertices must be in A. Such a vertex is

simplicial in G because none of its neighbors is in B. Furthermore, if G[A U S] is

complete, then any vertex of A is a simplicial vertex of G. Thus, in both cases, we see

that there exists at least one simplicial vertex in A. Using the same argument, we can see

that there exist also at least one simplicial vertex in B. Hence, as A is disconnected with

B in G\S, we see that if G is not a complete triangulated graph , it has at least 2 non-

adjacent simplicial vertices.

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Theorem 3.4 A graph G is triangulated iff it has a perfect vertex elimination scheme

Necessity: Let G be a triangulated graph. We shall prove this by induction. Assume that

every triangulated graph with fewer vertices than G has a perfect vertex elimination

scheme. By the previous lemma proved, since G is triangulated, G has a simplicial vertex

v. As G-v is still triangulated, G\v has a perfect vertex elimination scheme. Hence, by

induction hypothesis, v followed by a perfect scheme of G\v gives a perfect scheme of G.

Sufficiency: Let G have a perfect vertex elimination scheme {v1,v2, v3 … vn}. Consider a

cycle C of length greater than or equal to 4 in G. For any vertex vi in G that is contained

in C and i being the smallest suffix of all the vertices in C, vi is simplicial in the induced

subgraph of the set of vertices {vi, vi+1 … vn}. Thus, the neighbors of vi in C are adjacent

to one another. This, C has a chord and G is triangulated.

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4. MOPLEXES IN TRIANGULATED GRAPHS

Here we introduce a new term ‘moplex’.

4.1 Module

A module is a subset A of V (the vertex set of a graph) such that for all ai and aj in

A, N(ai) N(A) = N(aj) N(A) = N(A), i.e. every vertex of N(A) is adjacent to

every vertex in A.

A single vertex is a trivial module.

For a module that is a clique, all its neighbors are adjacent to every single vertex

in the clique itself.

4.2 Maximal clique module

A V is a maximal clique module if and only if A is both a module and a

clique, and A is maximal for both properties.

4.3 Moplexes

A moplex is a maximal clique module whose neighborhood is a minimal

separator.A moplex is simplicial iff its neighborhood is a clique, and it is trivial

iff it has only 1 vertex.

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Property 4.4 Every moplex M of a triangulated graph H is simplicial, and every

vertex of M is a simplicial vertex.

Let H be a triangulated graph and M be a moplex of H. By definition, N(M) is a minimal

separator. By Dirac’s characterization (lemma 2.3), N(M) is a clique. Hence, M is

simplicial, and every vertex in M is adjacent to every vertex in N(M).

For every vertex x in M, N(x) must be a clique. Hence, x is also simplicial.

Remark: The converse is not true. In a triangulated graph, a vertex can be simplicial

without belonging to any moplex.

In the graph below,

Minimal separators = {d, {b, c}}

Moplexes = {e, {f,g}}

Simplicial vertices = {e, f, g, a}

but a Moplex set

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Theorem 4.5 Any non-clique triangulated graph has at least 2 non-adjacent

simplicial moplexes.

Special case: When N = 3, the only connected non-clique graph is a P3 (path) of vertices,

in order, a, b and c. There are 3 moplexes in this graph; b is the minimal separator, but a

and c are 2 trivial moplexes.

Let G be a non-clique triangulated graph. Assume that the theorem is true for non-clique

triangulated graphs. Let S be a minimal separator of G which is a clique by Dirac’s

Theorem. Let also A and B be 2 full components of CC(S).

Case 1: If A S is a clique, N(A) = S. This implies that A is both a module and a clique.

For any x S, A {x} is not a module. For any y A S, A {y} is not a clique.

Therefore, A is a maximal clique module

Case 2: If A S is not a clique, by induction hypothesis, A S has 2 non-adjacent

moplexes. If each of these 2 moplexes are inclusive of vertices in both A and S, they will

be adjacent because S is a clique, which is a contradiction. Hence, one of the moplexes

(we call M) is contained in A. Thus, N(M) is a minimal separator in A S. This implies

that N(M) is also a minimal separator in G. Hence, M is a moplex in G. In either case,

there is simplicial moplex which is contained in A. Similarly, there is also such a moplex

contained in B.

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Theorem 4.6 A graph is triangulated iff one can repeatedly delete a simplicial

moplex (c.f. simplicial vertex) until the graph is a clique (i.e. there exists a ‘perfect

simplicial moplex elimination scheme’)

Necessity: Let G be a triangulated graph. There exist 2 non-adjacent simplicial moplexes

in G by theorem 3.5. Removing one of these 2 moplexes (call the removed moplex M),

G\M is still a triangulated graph. By continuously doing so, we will obtain a clique.

Sufficiency: Any vertex in M is simplicial by property 3.4. Hence, a simplicial moplex

elimination scheme is similar to a perfect vertex elimination scheme. By theorem 2.4, we

can conclude that every graph with a simplicial moplex elimination scheme is a

triangulated graph.

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5. GENERALIZATION OF DIRAC’S THEOREM TO ANY GRAPH

Lemma 5.1 Let H be a minimal triangulation of G and A be a moplex of H. Then

NH(A) = NG(A)

Let A be a moplex of H and a A. It is easily seen that NG(A) NH(A). Assume that

NG(A) NH(A). Consider a vertex z in NH(A) but not in NG(A). Since H is a minimal

triangulation of G, by the unique chord property, az is the unique chord of some 4-cycle

in H: axzya. However, since the neighborhood of A is a clique by definition, x must

already be adjacent to y for any x, y NH(A), and hence, az cannot be the unique chord.

Therefore, by contradiction, NG(A) = NH(A).

Lemma 5.2 Let H = (V, E + F) be a minimal triangulation of G = (V, E). If A is a

moplex of H, then A is a moplex of G.

Let A be a moplex of H. let N(A) be the neighbourhood of A. Note that N(A) = NG(A) =

NH(A). A N(A) is a clique of H. All we have to do now is to show that A is also a

moplex of G.

Suppose a, b A such that a NG(b). Then ab must belong to the minimal fill-in F, so

with the unique chord property, ab must be the unique chord of some 4-cycle axbya of H.

However, in H, x is adjacent to y since they are neighbors of a and A N(A) is a clique.

ab cannot be the unique chord of axbya. Hence, by contradiction, A is a clique of G.

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Assume A is not a module of G. z in N(A) such that z is not adjacent to a of A in G.

This edge az must then be in the minimal fill-in, which gives another contradiction

because of the unique chord property. Thus, A is a module of G.

If s N(A), A {s} is not a clique. If s N(A), s is adjacent to some vertex in B, where

B is the other full component of N(A); but the moplex containing A cannot be adjacent to

a B, which gives rise to a contradiction; Thus, A is maximal.

Theorem 5.3 Any non-clique graph G has at least 2 non-adjacent moplexes.

Let the triangulation of G be H. By theorem 3.5, H contains at least 2 non-adjacent

simplicial moplexes. By theorem 4.2 above, we know that a moplex of H is also a moplex

of G. Hence, we conclude that any non-clique graph has at least 2 non-adjacent

moplexes.

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6. REFERNCES

1. R. Balakrishnan, K. Ranganathan.1999.A textbook of Graph Theory. New York,

Springer

2. Anne Berry, J-P Bordat. 1996. Separability generalizes Dirac’s

3. Hans L. Bodlaender, Ton Kloks, Dieter Kratsch, Haiko Muller. 1998. Journal of

Graph Algorithms

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