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)
2
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
6.
TO ANY GRAPH
18
REFERENCES
21
5
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.
6
(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
10
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
12
components of G\S.Then S must contain x and yj for some j. By hypothesis, S is a 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 nonadjacent 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