This document discusses minimal vertex separators and perfect elimination orderings (PEOs) in chordal graphs. It begins by defining chordal graphs, PEOs, and vertex separators. It then discusses how to find minimal vertex separators (MVSs) using algorithms like LexBFS and how MVSs capture the structure of chordal graphs. The document also describes how to generate all PEOs of a chordal graph using a Hamilton cycle-based algorithm. It concludes by noting some properties and applications of chordal graphs, PEOs, and MVSs, such as using MVSs to decompose chordal graphs.

1. Minimal Vertex
Seperators & All PEOs
NAZLI TEMUR

2. 1 . C o r d a l G r a p h // D e f in it io n
2 .Pe r f e c t Elim in a t ion O r d e r in g //D e f in ition
3 .Ve r t e x Se p e r a t o r // D e f in it ion
4 .H o w t o f in d M VSs
5 .N e w Pr o o f o f L EXB F S v ia M VSs
6 .H o w t o f in d a ll PEO s
7 .O b s e r v a t io ns
O U T L I N E

3. C O R D A L G R A P H S
Definition. A Graph is chordal if it has no induced cycles larger than
riangles.
For a graph G on n vertices, the following conditions are equivalent:
G is chordal;
1.G has a perfect elimination ordering.
2. If every minimal vertex separator of a G is complete.
3. If every induced subgraph of G has a simplicial vertex.

4. PERFECT ELIMINATION ORDERINGS
A perfect elimination ordering in a graph is an ordering of the vertices of the graph
such that, for each vertex v, v and the neighbors of v that occur after v in the order,
form a clique.
A graph is chordal if and only if it has a perfect elimination ordering (Fulkerson & Gross 1965).
HIGHLIGHTS
A single perfect elimination ordering can be found in linear time using either the LexBFS algorithm or
MCS(Maximum cardinality search). //Rose, Lueker & Tarjan (1976) (see also Habib et al. 2000)
However neither of these algorithms can be used to proceed every PEO for a given chordal
graph.Habib et al. 2000

5. V E R T E X S E P E R A T O R S
• A separator is a set of vertices S, such that its removal increases the number of the
connected components in the graph,
• meaning // set of nodes S⊆G whose removal divides the graph to distinct [at least 2]
connected components.
• S is a minimal separator if no proper subset S′ of S is a separator in G.
• S is a minimal clique separator if it is a clique and a minimal separator.
According to a theorem of Dirac (1961), chordal graphs are graphs in which each minimal separator is
a clique; Dirac used this characterization to prove that chordal graphs are perfect.
HIGHLIGHTS

6. VERTEX SEPERATORS
EXAMPLE

7. D E T E R M I N I N G C O R D A L I T Y
In Cordal Graphs all MVSs induce a clique.
Counter example to show

8. DETERMINING CORDALITY
How Minimal Vertex Separator Captures the structure of Chordal Graphs
Structure of chordal graph
Since MVS of chordal graphs are cliques, the MVS of chordal graphs are K2-free.
The class of forbidden graphs are obtained by placing two vertices x and y on opposite sides of K2 and drawing paths of
arbitrary length from x to y through their vertices. This gives us the class of forbidden induced subgraphs for chordal
graphs. Minimality requires these paths to be disjoint except at x and y. So for chordal graphs our class of minimal
forbidden graphs are the cycles Cn, n ≥ 4. Our first step is to find similar results for graphs whose MVS are C-free, for C ∈
C. In [5], Aboulker et al. found forbidden induced subgraphs for graphs where every induced subgraph has a vertex v
whose neighbourhood N(v) is F-free, for a set of graphs F. Not surprisingly the forbidden induced subgraphs we got, as
well as in [5], are Truemper configurations or their close relatives. Truemper configurations play a key role in understanding
the structures of perfect graphs.
Another advantage of chordal graphs is that every such graph has a clique decomposition. Hence by iteratively adding a
vertex joined to a clique, we can construct chordal graphs from a single vertex. These kinds of construction algo- rithms
lead to fast algorithms for computations on graphs in this class. In [7], Trotignon and Vuˇskovi ́c develop such construction

9. R E L A T I O N P E O - M V S
PROPERTIES
Concluding an ordering is PEO via Minimal Vertex Seperators

10. NEW PROOF OF LEXBFS VIA MVS

12. H O W T O G E N E R A T E M V S

13. K L B - L G B A L G O R I T H M S
There are several algorithms to compute MVSs, such as KLB , LGB , F Ruskey., L. Sunil Chandran1
,
Fabrizio Grandoni ,Kumar and Madhavan and so on … Some of them are using initial PEOs that are
generated via specific algorithms such as MCS or clique trees, some of them does not require computation
of PEOs.
Kernighan-Lin algorithm as an edge separator minimizing bisection algorithm can be seen to find its niche
in refining results obtained via a versatility of algorithms, In other words with the fact that constructing
vertex separators from edge separators which is not always effective.
Line Graph Bisection algorithm has been inspired from the well-known Kernighan-Lin [9] graph bisection
algorithm (KLB) by incorporating some novel modifications so as to make finding vertex separators
possible. KLB finds small edge separators in Ο(n 2 log n) time where n denotes the number
of nodes in the graph. Fiduccia and Mattheyses [11] improved this running time to Ο( E )

14. L E X B F S
For a graph G= (V,E) , a Lexicographic Breadth First Search of G, denoted LExBFS(G) , is a breadth first
search in which preference is given to vertices whose neighbors have been visited earliest in the search //
LexBFS
• This is a significantly simplified implementation of Lex-BFS which has come to be known as the MCS
(maximum cardinality search) algorithm.
• Knowing specific processed neighbors (i.e. labels) is not necessary; only need is to maintain and
compare the cardinality of processed neighbors.
M C S

15. MOPLEXES IN CHORDAL 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.
Property 4.4 Every moplex M of a chordal 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.

16. MOPLEXES IN CHORDAL GRAPHS
Minimal separators = {d, {b, c}}
Moplexes = {e, {f,g}}
Simplicial vertices = {e, f, g, a}
but a ∉ Moplex set
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.
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), GM is still a triangulated graph. By continuously doing so, we will obtain a clique.

17. G E N E R A T I N G M V S V I A L E X B F S
A simple and optimal process which generates the minimal separators and maximal cliques of a chordal
graph in a single pass of either LexBFS or MCS, without requiring the preliminary computation of a PEO is
exist.
Though both LexBFS and MCS yield an optimal linear-time process for this problem, it is important to note
that they define a different set of PEO of a chordal graph, and exhibit different local behaviors.
It may be important to use one or the other, depending on the intended application.

18. G E N E R A T I N G M V S V I A L E X B F S
LexBFS defines a moplex ordering and an associated peo: LexBFS always numbers as 1 a vertex belonging to a
moplex (which we will call X1). They also proved that the vertices of X1 receive consecutive numbers by LexBFS.
These properties are true at each step of LexBFS in the transitory elimination graph. Therefore, LexBFS defines a
moplex elimination (X1,...,Xk), by numbering consecutively the vertices of X1, then numbering consecutively the
vertices of X2, and so forth:
Theorem 3.11 In a chordal graph, LexBFS defines a moplex ordering. Note that it is easy to deduce that MCS also
defines a moplex ordering.

19. G E N E R A T I N G M V S V I A M C S

20. H O W T O G E N E R A T E A L L P E O S

21. G E N E R A T I N G A L L P E O S V I A H A M I L T O N
C L + P E O
Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once.
Whether two adjacent elements of a PEO can be swapped to obtain a new PEO was the question
that inspire this algorithmm.
This leads first algorithm for generating all PEOs of a chordal graph G in constant amortized
time.[Initialization of the algorithm can be performed in linear time using clique trees]
LEXBFS and MCS can not be used the obtain primary PEO because for this algorithm is it required
to remove pairs or simlicial vertices that is not valid for Lbfs and Mcs
There are two characteristics of Perfect Elimination Orderings , One is Chordless Path the Other
one is Minimal Vertex Seperators.
When authors were searching for a PEO ordering to be able to initiate the next algorithm they
identified these characteristics. They disprove a claim that is done by Simon . The claim was for
some PEOs generated by LEXBFS and MCS algorithm result is not hold. But via chordless cycle
and MVSs author conclude that results true for all PEOs.

22. G E N E R A T I N G M V S V I A H A M I L T O N
C L + P E O
PEOs of a chordal graph G form the basic words of antimatroid language. So author takes the advantage of
an algorithm that is called GrayCode Algorithm.
The basic idea behind is to traverse a particular Hamilton Cycle in the Graph. Such a traversal will visit all the
perfect elimination orderings twice.
In the new algorithm author prints only every second visited PEO in Hamilton Cycle.

23. G E N E R A T I N G M V S V I A H A M I L T O N
C L + P E O
Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once.
F. Ruskeyc;∗;2, J. Sawada,

24. OBSERVATIONS
Chordal Graphs
• Every chordal graph G has a simplicial vertex. If G is not a complete graph, then it has
two simplicial vertices that are not adjacent.
• If G is chordal, then after deleting some vertices, the remaining graph is still chordal.
{Simplicial Node Property}
• So in order to show that every chordal graph has a perfect elimination order, it suffices to
show that every chordal has a simplicial vertex;

25. OBSERVATIONS
Perfect Elimination Ordering
Another application of perfect elimination orderings is finding a maximum clique of a chordal graph in
polynomial-time, while the same problem for general graphs is NP-complete.
More generally, a chordal graph can have only linearly many maximal cliques, while non-chordal graphs may
have exponentially many.
To list all maximal cliques of a chordal graph, simply find a perfect elimination ordering, form a clique for each
vertex v together with the neighbors of v that are later than v in the perfect elimination ordering, and test
whether each of the resulting cliques is maximal.Sensors)

26. OBSERVATIONS
Minimum Vertex Separators
A separator S with two roughly equal-sized sets A and B has the desirable effect of dividing the independent work
evenly between two processors. Moreover, a small separator implies that the remaining work load in computing S is
relatively small. A recursive use of separators can provide a framework suitable for parallelization using more than two
processors.
A subset S⊆V is called as u-v separator of G in G-S.The vertices u-v are in two different connected components.A u-v
separator is called a minimal u-v separator for some u-v
The set of minimal vertex separators constitute a unique separators of a chordal graph and capture the structure of the graph.
So we have the following hereditary properties: for MINIMAL VERTEX SEPARATOR
• MVS is K2-free, i.e. MVS induces a clique. This gives the class of chordal graphs.
• MVS is K2- free, i.e. MVS induces an independent set. This gives the class of graphs with no cycle with a unique chord.
• MVS is K3-free. This gives the class of graphs with MVS Gxy such that the independence number α(Gxy) ≤ 2.
• MVS is K1 ∪ K2-free, i.e. MVS induces a complete multipartite graph.

27. OBSERVATIONS
Minimum Vertex Separators
• An undirected graph G is Chordal if and only if every minimal vertex separator of it induces a clique.
• The problem of listing all minimal separators is one of the fundamental enumeration problems in graph theory,
which has great practical importance in reliability analysis for networks and operations research for scheduling
problems..
A linear time algorithm to list the minimal separators of chordal graphs􏰻 // L. Sunil Chandran1, Fabrizio
Grandoni
• Identification of the set of minimal vertex separators of a chordal graph enables us to decompose the graph into
subgraphs that are again chordal.
• These classes would be useful in gaining insights into the nature of problems that are hard for classes of
chordal graphs. We can restrict problems into the new sub class of the Chordal graphs and study the
behaviour. // Ex: k-seperator chordal(all the minimal vertex separators are exactlysize of k.) For the Class of 2

28. OBSERVATIONS
Minimum Vertex Separators
• A structure that generalizes the clique tree notation called the ‘reduced clique hyper graph’ .Minimal
separators are useful in characterizing the edges of rch and algorithmic characterization of these
mvss’ leads to an efficient algorithm for constructing rch of Chordal Graphs.Then process can be
continued until the sub graphs are separator -free chordal graphs,namely cliques.
• According to a theorem of Dirac (1961), chordal graphs are graphs in which each minimal separator
is a clique; Dirac used this characterization to prove that chordal graphs are perfect.
• The size and multiplicity of minimal vertex separators are two parameters on which if we impose
conditions we can obtain several different subclasses of chordal graphs.
• New proof of LexBFS is provided wrt Minimal Vertex Separators in 1994 Klaus Simon

29. D E M O N S T R A T I O N
S

30. Here we re going to
visualize vertex
separators.
{2,3,4} {2,4}{2,4,5}{1,2,4}
DEMO

31. H e r e w e r e g o i n g to
v i s u a l i z e m i n i m a l
v e r te x s e p a r a to rs.
{ 2 ,4 }
DEMO

32. R E F E R A N C E S
For MCS
http://pgm.stanford.edu/Algs/page-312.pdf
http://www.ii.uib.no/~pinar/MCS-M.pdf
http://www.cs.upc.edu/~valiente/graph-00-01-e.pdf
For Perfect Elimination Orderings
http://www.cis.uoguelph.ca/~sawada/papers/chordal.pdf
https://www.math.binghamton.edu/zaslav/Oldcourses/580.S13/bartlett.MC2011_perfectgraphs_wk1_day3.pdf
https://books.google.fr/books?id=8bjSBQAAQBAJ&pg=PA87&lpg=PA87&dq=Alan+Hoffman+perfect+elimination+or
dering&source=bl&ots=UvM-wUMXF7&sig=HsP83gL-
ju07NlJfpiLQgJyrEXA&hl=tr&sa=X&ei=WQNOVdrII8azUf7agMAM&ved=0CCkQ6AEwAQ#v=onepage&q=Alan%20
Hoffman%20perfect%20elimination%20ordering&f=false
Cordal Graph and Clique Trees
https://www.math.binghamton.edu/zaslav/Oldcourses/580.S13/blair-peyton.chordal-graphs-clique-
trees.ornl1992.pdf
cordal and their clique graphs
http://www.liafa.jussieu.fr/~habib/Documents/WGChordaux.pdf

33. R E F E R A N C E S
Cordal Graphs ..
Thesis from University of Singapore
Properties of Chordal Graphs
Graphs
http://www.math.binghamton.edu/zaslav/581.F14/course-notes-chapter1.pdf
Vertex Seperator
http://en.wikipedia.org/wiki/Vertex_separator
http://www.ncbi.nlm.nih.gov/pmc/articles/PMC3927507/f
Minimal vertex seperator
http://www.ii.uib.no/~pinar/MinTriSurvey.pdf
https://books.google.fr/books?id=FKeoCAAAQBAJ&pg=PA355&lpg=PA355&dq=a.+parra+and+p+scheffler+how+to+use+minimal+separat
ors+of+a+chordal+graph+for+its+chordal+triangulation&source=bl&ots=oKtjp-
kRC8&sig=j0o_Ir1_cOeAPrAp1glYf2DFy_E&hl=tr&sa=X&ei=SDlPVe2SOIaAUa3sgNAB&ved=0CDQQ6AEwAg#v=onepage&q=a.%20parr
a%20and%20p%20scheffler%20how%20to%20use%20minimal%20separators%20of%20a%20chordal%20graph%20for%20its%20chord
al%20triangulation&f=false
http://www.mate.unlp.edu.ar/~pdecaria/57185.pdf not useful
Generating all minimal vertex seperator
http://www.isima.fr/~berry/generating.ps
Minimal Stable Vertex Seperator
http://arxiv.org/pdf/1103.2913.pdf
Minimal Seperator at Cordal Graph
http://people.idsia.ch/~grandoni/Pubblicazioni/CG06dm.pdf
http://ac.els-cdn.com/S0166218X98001231/1-s2.0-S0166218X98001231-main.pdf?_tid=51b34232-f708-11e4-86a2-
00000aab0f27&acdnat=1431257763_981ba2c5eac5bced0f5524a9b8e1a8f9
http://www.sciencedirect.com/science/article/pii/S0166218X10001824

34. R E F E R A N C E S
A linear time algorithm to list the minimal separators of chordal graphs􏰻
L. Sunil Chandran1, Fabrizio Grandoni http://www.sciencedirect.com/science/article/pii/S0166218X10001824
Important Source includes LEXBFS
Lexicographic Breadth First for Chordal Graphs Klaus Simon
http://citeseerx.ist.psu.edu/viewdoc/download?doi=10.1.1.56.5018&rep=rep1&type=ps
http://www.isima.fr/~berry/RR-10-04
http://www.crraoaimscs.org/researchreport/Downloads/2014/RR2014-15.pdf
https://books.google.fr/books?id=GbsPBwAAQBAJ&pg=PA225&lpg=PA225&dq=minimal+vertex+separator+lexbfs&source=bl&ots=gK2RH1krfc&sig
=pt-
VraENVg3F1cxsCQcEwtsg7k8&hl=tr&sa=X&ei=LUZPVY7ZCMLpUuv1gKAE&ved=0CE0Q6AEwBQ#v=onepage&q=minimal%20vertex%20separato
r%20lexbfs&f=false
https://books.google.fr/books?id=oDrTFgWtLdgC&pg=PA141&lpg=PA141&dq=confluence+point+of+C+graph&source=bl&ots=rMlrlenJuY&sig=4lVjp
cXoPyQiVz_rjKSgx5w_rcw&hl=tr&sa=X&ei=WG1PVYLJOcv9UMLagOgM&ved=0CCIQ6AEwAA#v=onepage&q=confluence%20point%20of%20C%
20graph&f=false
http://i.stanford.edu/pub/cstr/reports/cs/tr/75/531/CS-TR-75-531.pdf
http://www2.ii.uib.no/~pinar/MCSM-r.pdf
http://www.ics.uci.edu/~agelfand/pub/Post-ProcessingtoReduceInducedWidth.pdf
PEO Structure..
http://www.cs.upc.edu/~valiente/graph-00-01-e.pdf

35. – N A Z L I T E M U R
“Thank you for listening.”