Min ima l Ver tex Sepera to rs amp A l l PEOs

NAZLI TEMUR

1 Corda l Graph Defi ni t ion2Perfect E l iminat ion Order ing Defi nit ion

3Vertex Seperator Defi ni t ion4How to fi nd MVSs5New Proof of LEXBFS v ia MVSs6How to fi nd a l l PEOs 7 Observat ions

OUTLINE

CORDAL GRAPHS

Definition A Graph is chordal if it has no induced cycles larger than triangles

For a graph G on n vertices the following conditions are equivalent

G is chordal

1G 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

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 amp 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 amp 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

1 Corda l Graph Defi ni t ion2Perfect E l iminat ion Order ing Defi nit ion

3Vertex Seperator Defi ni t ion4How to fi nd MVSs5New Proof of LEXBFS v ia MVSs6How to fi nd a l l PEOs 7 Observat ions

OUTLINE

CORDAL GRAPHS

Definition A Graph is chordal if it has no induced cycles larger than triangles

For a graph G on n vertices the following conditions are equivalent

G is chordal

1G 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

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 amp 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 amp 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

CORDAL GRAPHS

Definition A Graph is chordal if it has no induced cycles larger than triangles

For a graph G on n vertices the following conditions are equivalent

G is chordal

1G 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

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 amp 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 amp 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

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 amp 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 amp 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

VERTEX SEPERATORS

bull A separator is a set of vertices S such that its removal increases the number of the connected components in the graph

bull meaning set of nodes SsubeG whose removal divides the graph to distinct [at least 2] connected components

bullS is a minimal separator if no proper subset Sprime of S is a separator in G bullS 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

VERTEX SEPERATORSEXAMPLE

DETERMINING CORDALITY

In Cordal Graphs all MVSs induce a clique

Counter example to show

DETERMINING CORDALITYHow 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 ge 4 Our first step is to find

similar results for graphs whose MVS are C-free for C isin 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 Mc develop such construction techniques for graphs with no cycle with a unique

chord They give a structural definition for graphs with no cycles with a unique chord and present

polynomial algorithms for recognition (of order O(nm)) finding clique number (of order O(n + m)) and

chromatic number (of order O(nm)) for such graphs They also prove that finding a maximal stable set for a

graph in this class is NP-Complete

DETERMINING CORDALITY

In Cordal Graphs all MVSs induce a clique

Counter example to show

DETERMINING CORDALITYHow 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 ge 4 Our first step is to find

similar results for graphs whose MVS are C-free for C isin 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 Mc develop such construction techniques for graphs with no cycle with a unique

chord They give a structural definition for graphs with no cycles with a unique chord and present

polynomial algorithms for recognition (of order O(nm)) finding clique number (of order O(n + m)) and

chromatic number (of order O(nm)) for such graphs They also prove that finding a maximal stable set for a

graph in this class is NP-Complete

DETERMINING CORDALITYHow 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 ge 4 Our first step is to find

similar results for graphs whose MVS are C-free for C isin 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 Mc develop such construction techniques for graphs with no cycle with a unique

chord They give a structural definition for graphs with no cycles with a unique chord and present

polynomial algorithms for recognition (of order O(nm)) finding clique number (of order O(n + m)) and

chromatic number (of order O(nm)) for such graphs They also prove that finding a maximal stable set for a

graph in this class is NP-Complete

RELATION PEO-MVS

PROPERTIES

Concluding an ordering is PEO via Minimal Vertex Seperators

NEW PROOF OF LEXBFS VIA MVS

HOW TO GENERATE MVS

KLB - LGB ALGORITHMS

There are several algorithms to compute MVSs such as KLB LGB F Ruskey L Sunil Chandran1 Fabrizio Grandoni Kumar and Madhavan and so on hellip 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 )

LEXBFS

For a graph G= (VE) 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

bull This is a significantly simplified implementation of Lex-BFS which has come to be known as the MCS (maximum cardinality search) algorithm

bull Knowing specific processed neighbors (ie labels) is not necessary only need is to maintain and compare the cardinality of processed neighbors

MCS

NEW PROOF OF LEXBFS VIA MVS

HOW TO GENERATE MVS

KLB - LGB ALGORITHMS

There are several algorithms to compute MVSs such as KLB LGB F Ruskey L Sunil Chandran1 Fabrizio Grandoni Kumar and Madhavan and so on hellip 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 )

LEXBFS

For a graph G= (VE) 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

bull This is a significantly simplified implementation of Lex-BFS which has come to be known as the MCS (maximum cardinality search) algorithm

bull Knowing specific processed neighbors (ie labels) is not necessary only need is to maintain and compare the cardinality of processed neighbors

MCS

HOW TO GENERATE MVS

KLB - LGB ALGORITHMS

There are several algorithms to compute MVSs such as KLB LGB F Ruskey L Sunil Chandran1 Fabrizio Grandoni Kumar and Madhavan and so on hellip 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 )

LEXBFS

For a graph G= (VE) 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

bull This is a significantly simplified implementation of Lex-BFS which has come to be known as the MCS (maximum cardinality search) algorithm

bull Knowing specific processed neighbors (ie labels) is not necessary only need is to maintain and compare the cardinality of processed neighbors

MCS

KLB - LGB ALGORITHMS

There are several algorithms to compute MVSs such as KLB LGB F Ruskey L Sunil Chandran1 Fabrizio Grandoni Kumar and Madhavan and so on hellip 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 )

LEXBFS

For a graph G= (VE) 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

bull This is a significantly simplified implementation of Lex-BFS which has come to be known as the MCS (maximum cardinality search) algorithm

bull Knowing specific processed neighbors (ie labels) is not necessary only need is to maintain and compare the cardinality of processed neighbors

MCS

LEXBFS

For a graph G= (VE) 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

bull This is a significantly simplified implementation of Lex-BFS which has come to be known as the MCS (maximum cardinality search) algorithm

bull Knowing specific processed neighbors (ie labels) is not necessary only need is to maintain and compare the cardinality of processed neighbors

MCS

MOPLEXES IN CHORDAL GRAPHS

Here we introduce a new term lsquomoplexrsquo 41 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) cap N(A) = N(aj) cap N(A) = N(A) ie every vertex of N(A) is adjacent to every vertex in A

A single vertex is a trivial moduleFor a module that is a clique all its neighbors are adjacent to every single vertex in the clique

itself

42 Maximal clique moduleA sub 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

43 MoplexesA 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 44 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 Diracrsquos characterization (lemma 23) 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

MOPLEXES IN CHORDAL GRAPHS

Minimal separators = d b cMoplexes = e fgSimplicial vertices = e f g abut a notin Moplex set

Theorem 45 Any non-clique triangulated graph has at least 2 non-adjacent simplicial moplexesSpecial 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 moplexesLet 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 Diracrsquos Theorem Let also A and B be 2 full components of CC(S)Case 1 If A cup S is a clique N(A) = S This implies that A is both a module and a clique For any x isin S A cup x is not a module For any y notin A cup S A cup y is not a clique Therefore A is a maximal clique moduleCase 2 If A cup S is not a clique by induction hypothesis A cup 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 cup 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 46 A graph is triangulated iff one can repeatedly delete a simplicial moplex (cf simplicial vertex) until the graph is a clique (ie there exists a lsquoperfect simplicial moplex elimination schemersquo)Necessity Let G be a triangulated graph There exist 2 non-adjacent simplicial moplexes in G by theorem 35 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

Sufficiency Any vertex in M is simplicial by property 34 Hence a simplicial moplex elimination scheme is similar to

a perfect vertex elimination scheme By theorem 24 we can conclude that every graph with a simplicial moplex

elimination scheme is a triangulated graph

GENERATING MVS V IA LEXBFSA 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

GENERATING MVS V IA LEXBFS

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 (X1Xk) by numbering consecutively the vertices of X1 then numbering consecutively the vertices of X2 and so forth

Theorem 311 In a chordal graph LexBFS defines a moplex ordering Note that it is easy to deduce that MCS also defines a moplex ordering

GENERATING MVS V IA MCS

HOW TO GENERATE ALL PEOS

GENERATING ALLPEOS V IA HAMILTON CL+PEOHamilton 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

GENERATING MVS V IA HAMILTON CL+PEO

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

GENERATING MVS V IA HAMILTON CL+PEO

Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once

F Ruskeyclowast2 J Sawada

MOPLEXES IN CHORDAL GRAPHS

Minimal separators = d b cMoplexes = e fgSimplicial vertices = e f g abut a notin Moplex set

Theorem 45 Any non-clique triangulated graph has at least 2 non-adjacent simplicial moplexesSpecial 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 moplexesLet 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 Diracrsquos Theorem Let also A and B be 2 full components of CC(S)Case 1 If A cup S is a clique N(A) = S This implies that A is both a module and a clique For any x isin S A cup x is not a module For any y notin A cup S A cup y is not a clique Therefore A is a maximal clique moduleCase 2 If A cup S is not a clique by induction hypothesis A cup 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 cup 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 46 A graph is triangulated iff one can repeatedly delete a simplicial moplex (cf simplicial vertex) until the graph is a clique (ie there exists a lsquoperfect simplicial moplex elimination schemersquo)Necessity Let G be a triangulated graph There exist 2 non-adjacent simplicial moplexes in G by theorem 35 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

Sufficiency Any vertex in M is simplicial by property 34 Hence a simplicial moplex elimination scheme is similar to

a perfect vertex elimination scheme By theorem 24 we can conclude that every graph with a simplicial moplex

elimination scheme is a triangulated graph

GENERATING MVS V IA LEXBFSA 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

GENERATING MVS V IA LEXBFS

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 (X1Xk) by numbering consecutively the vertices of X1 then numbering consecutively the vertices of X2 and so forth

Theorem 311 In a chordal graph LexBFS defines a moplex ordering Note that it is easy to deduce that MCS also defines a moplex ordering

GENERATING MVS V IA MCS

HOW TO GENERATE ALL PEOS

GENERATING ALLPEOS V IA HAMILTON CL+PEOHamilton 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

GENERATING MVS V IA HAMILTON CL+PEO

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

GENERATING MVS V IA HAMILTON CL+PEO

Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once

F Ruskeyclowast2 J Sawada

GENERATING MVS V IA LEXBFSA 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

GENERATING MVS V IA LEXBFS

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 (X1Xk) by numbering consecutively the vertices of X1 then numbering consecutively the vertices of X2 and so forth

Theorem 311 In a chordal graph LexBFS defines a moplex ordering Note that it is easy to deduce that MCS also defines a moplex ordering

GENERATING MVS V IA MCS

HOW TO GENERATE ALL PEOS

GENERATING ALLPEOS V IA HAMILTON CL+PEOHamilton 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

GENERATING MVS V IA HAMILTON CL+PEO

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

GENERATING MVS V IA HAMILTON CL+PEO

Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once

F Ruskeyclowast2 J Sawada

GENERATING MVS V IA LEXBFS

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 (X1Xk) by numbering consecutively the vertices of X1 then numbering consecutively the vertices of X2 and so forth

Theorem 311 In a chordal graph LexBFS defines a moplex ordering Note that it is easy to deduce that MCS also defines a moplex ordering

GENERATING MVS V IA MCS

HOW TO GENERATE ALL PEOS

GENERATING ALLPEOS V IA HAMILTON CL+PEOHamilton 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

GENERATING MVS V IA HAMILTON CL+PEO

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

GENERATING MVS V IA HAMILTON CL+PEO

Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once

F Ruskeyclowast2 J Sawada

GENERATING MVS V IA MCS

HOW TO GENERATE ALL PEOS

GENERATING ALLPEOS V IA HAMILTON CL+PEOHamilton 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

GENERATING MVS V IA HAMILTON CL+PEO

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

GENERATING MVS V IA HAMILTON CL+PEO

Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once

F Ruskeyclowast2 J Sawada

HOW TO GENERATE ALL PEOS

GENERATING ALLPEOS V IA HAMILTON CL+PEOHamilton 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

GENERATING MVS V IA HAMILTON CL+PEO

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

GENERATING MVS V IA HAMILTON CL+PEO

Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once

F Ruskeyclowast2 J Sawada

GENERATING ALLPEOS V IA HAMILTON CL+PEOHamilton 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

GENERATING MVS V IA HAMILTON CL+PEO

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

GENERATING MVS V IA HAMILTON CL+PEO

Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once

F Ruskeyclowast2 J Sawada

GENERATING MVS V IA HAMILTON CL+PEO

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

GENERATING MVS V IA HAMILTON CL+PEO

Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once

F Ruskeyclowast2 J Sawada

GENERATING MVS V IA HAMILTON CL+PEO

Hamilton cycle is a path in an undirected or directed graph that visits each vertex exactly once

F Ruskeyclowast2 J Sawada

OBSERVATIONSChordal Graphs

bull 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

bull If G is chordal then after deleting some vertices the remaining graph is still chordal Simplicial Node Property

bull 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

OBSERVATIONSPerfect 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 maximalSensors)

OBSERVATIONSPerfect 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 maximalSensors)

OBSERVATIONSMinimum 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 SsubeV is called as u-v separator of G in G-SThe vertices u-v are in two different connected

componentsA 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

bull MVS is K2-free ie MVS induces a clique This gives the class of chordal graphs

bull MVS is K2- free ie MVS induces an independent set This gives the class of graphs with no cycle with a unique chord

bull MVS is K3-free This gives the class of graphs with MVS Gxy such that the independence number α(Gxy) le 2

bull MVS is K1 cup K2-free ie MVS induces a complete multipartite graph

bull MVS is P3-free ie MVS induces a collection of cliques

bull MVS is K3-free ie MVS is ∆-free

OBSERVATIONSMinimum Vertex Separators

bull An undirected graph G is Chordal if and only if every minimal vertex separator of it induces a

clique

bull 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 graphs1113147 L Sunil Chandran1

Fabrizio Grandoni

bull Identification of the set of minimal vertex separators of a chordal graph enables us to decompose

the graph into subgraphs that are again chordal

bull 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 sep chordal properly contains 2 trees and Hamilton Circuit

Problem can be efficiently solvedWhereas it remains np complete for 3-sept chordal

OBSERVATIONSMinimum Vertex Separators

bull A structure that generalizes the clique tree notation called the lsquoreduced clique hyper

graphrsquo Minimal separators are useful in characterizing the edges of rch and algorithmic

characterization of these mvssrsquo leads to an efficient algorithm for constructing rch of

Chordal GraphsThen process can be continued until the sub graphs are separator -free

chordal graphsnamely cliques

bull 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

bull 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

bull New proof of LexBFS is provided wrt Minimal Vertex Separators in 1994 Klaus Simon

D E M O N S T RAT I O N S

Here we re go ing to v i sua l i ze ver tex separa to rs

2 3 4 2 42 4 51 2 4

DEMO

Here we re go ing to v i sua l i ze m in ima l ve r tex sepa ra to r s

2 4

DEMO

REFERANCES

For MCShttppgmstanfordeduAlgspage-312pdfhttpwwwiiuibno~pinarMCS-Mpdfhttpwwwcsupcedu~valientegraph-00-01-epdf

For Perfect Elimination Orderingshttpwwwcisuoguelphca~sawadapaperschordalpdfhttpswwwmathbinghamtoneduzaslavOldcourses580S13bartlettMC2011_perfectgraphs_wk1_day3pdf

httpsbooksgooglefrbooksid=8bjSBQAAQBAJamppg=PA87amplpg=PA87ampdq=Alan+Hoffman+perfect+elimination+orderingampsource=blampots=UvM-wUMXF7ampsig=HsP83gL-ju07NlJfpiLQgJyrEXAamphl=trampsa=Xampei=WQNOVdrII8azUf7agMAMampved=0CCkQ6AEwAQv=onepageampq=Alan20Hoffman20perfect20elimination20orderingampf=false

Cordal Graph and Clique Trees

httpswwwmathbinghamtoneduzaslavOldcourses580S13blair-peytonchordal-graphs-clique-treesornl1992pdfcordal and their clique graphshttpwwwliafajussieufr~habibDocumentsWGChordauxpdf

OBSERVATIONSMinimum Vertex Separators

bull An undirected graph G is Chordal if and only if every minimal vertex separator of it induces a

clique

bull 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 graphs1113147 L Sunil Chandran1

Fabrizio Grandoni

bull Identification of the set of minimal vertex separators of a chordal graph enables us to decompose

the graph into subgraphs that are again chordal

bull 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 sep chordal properly contains 2 trees and Hamilton Circuit

Problem can be efficiently solvedWhereas it remains np complete for 3-sept chordal

OBSERVATIONSMinimum Vertex Separators

bull A structure that generalizes the clique tree notation called the lsquoreduced clique hyper

graphrsquo Minimal separators are useful in characterizing the edges of rch and algorithmic

characterization of these mvssrsquo leads to an efficient algorithm for constructing rch of

Chordal GraphsThen process can be continued until the sub graphs are separator -free

chordal graphsnamely cliques

bull 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

bull 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

bull New proof of LexBFS is provided wrt Minimal Vertex Separators in 1994 Klaus Simon

D E M O N S T RAT I O N S

Here we re go ing to v i sua l i ze ver tex separa to rs

2 3 4 2 42 4 51 2 4

DEMO

Here we re go ing to v i sua l i ze m in ima l ve r tex sepa ra to r s

2 4

DEMO

REFERANCES

For MCShttppgmstanfordeduAlgspage-312pdfhttpwwwiiuibno~pinarMCS-Mpdfhttpwwwcsupcedu~valientegraph-00-01-epdf

For Perfect Elimination Orderingshttpwwwcisuoguelphca~sawadapaperschordalpdfhttpswwwmathbinghamtoneduzaslavOldcourses580S13bartlettMC2011_perfectgraphs_wk1_day3pdf

httpsbooksgooglefrbooksid=8bjSBQAAQBAJamppg=PA87amplpg=PA87ampdq=Alan+Hoffman+perfect+elimination+orderingampsource=blampots=UvM-wUMXF7ampsig=HsP83gL-ju07NlJfpiLQgJyrEXAamphl=trampsa=Xampei=WQNOVdrII8azUf7agMAMampved=0CCkQ6AEwAQv=onepageampq=Alan20Hoffman20perfect20elimination20orderingampf=false

Cordal Graph and Clique Trees

httpswwwmathbinghamtoneduzaslavOldcourses580S13blair-peytonchordal-graphs-clique-treesornl1992pdfcordal and their clique graphshttpwwwliafajussieufr~habibDocumentsWGChordauxpdf

OBSERVATIONSMinimum Vertex Separators

bull A structure that generalizes the clique tree notation called the lsquoreduced clique hyper

graphrsquo Minimal separators are useful in characterizing the edges of rch and algorithmic

characterization of these mvssrsquo leads to an efficient algorithm for constructing rch of

Chordal GraphsThen process can be continued until the sub graphs are separator -free

chordal graphsnamely cliques

bull 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

bull 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

bull New proof of LexBFS is provided wrt Minimal Vertex Separators in 1994 Klaus Simon

D E M O N S T RAT I O N S

Here we re go ing to v i sua l i ze ver tex separa to rs

2 3 4 2 42 4 51 2 4

DEMO

Here we re go ing to v i sua l i ze m in ima l ve r tex sepa ra to r s

2 4

DEMO

REFERANCES

For MCShttppgmstanfordeduAlgspage-312pdfhttpwwwiiuibno~pinarMCS-Mpdfhttpwwwcsupcedu~valientegraph-00-01-epdf

For Perfect Elimination Orderingshttpwwwcisuoguelphca~sawadapaperschordalpdfhttpswwwmathbinghamtoneduzaslavOldcourses580S13bartlettMC2011_perfectgraphs_wk1_day3pdf

httpsbooksgooglefrbooksid=8bjSBQAAQBAJamppg=PA87amplpg=PA87ampdq=Alan+Hoffman+perfect+elimination+orderingampsource=blampots=UvM-wUMXF7ampsig=HsP83gL-ju07NlJfpiLQgJyrEXAamphl=trampsa=Xampei=WQNOVdrII8azUf7agMAMampved=0CCkQ6AEwAQv=onepageampq=Alan20Hoffman20perfect20elimination20orderingampf=false

Cordal Graph and Clique Trees

httpswwwmathbinghamtoneduzaslavOldcourses580S13blair-peytonchordal-graphs-clique-treesornl1992pdfcordal and their clique graphshttpwwwliafajussieufr~habibDocumentsWGChordauxpdf

D E M O N S T RAT I O N S

Here we re go ing to v i sua l i ze ver tex separa to rs

2 3 4 2 42 4 51 2 4

DEMO

Here we re go ing to v i sua l i ze m in ima l ve r tex sepa ra to r s

2 4

DEMO

REFERANCES

For MCShttppgmstanfordeduAlgspage-312pdfhttpwwwiiuibno~pinarMCS-Mpdfhttpwwwcsupcedu~valientegraph-00-01-epdf

For Perfect Elimination Orderingshttpwwwcisuoguelphca~sawadapaperschordalpdfhttpswwwmathbinghamtoneduzaslavOldcourses580S13bartlettMC2011_perfectgraphs_wk1_day3pdf

httpsbooksgooglefrbooksid=8bjSBQAAQBAJamppg=PA87amplpg=PA87ampdq=Alan+Hoffman+perfect+elimination+orderingampsource=blampots=UvM-wUMXF7ampsig=HsP83gL-ju07NlJfpiLQgJyrEXAamphl=trampsa=Xampei=WQNOVdrII8azUf7agMAMampved=0CCkQ6AEwAQv=onepageampq=Alan20Hoffman20perfect20elimination20orderingampf=false

Cordal Graph and Clique Trees

httpswwwmathbinghamtoneduzaslavOldcourses580S13blair-peytonchordal-graphs-clique-treesornl1992pdfcordal and their clique graphshttpwwwliafajussieufr~habibDocumentsWGChordauxpdf

Here we re go ing to v i sua l i ze ver tex separa to rs

2 3 4 2 42 4 51 2 4

DEMO

Here we re go ing to v i sua l i ze m in ima l ve r tex sepa ra to r s

2 4

DEMO

REFERANCES

For MCShttppgmstanfordeduAlgspage-312pdfhttpwwwiiuibno~pinarMCS-Mpdfhttpwwwcsupcedu~valientegraph-00-01-epdf

For Perfect Elimination Orderingshttpwwwcisuoguelphca~sawadapaperschordalpdfhttpswwwmathbinghamtoneduzaslavOldcourses580S13bartlettMC2011_perfectgraphs_wk1_day3pdf

httpsbooksgooglefrbooksid=8bjSBQAAQBAJamppg=PA87amplpg=PA87ampdq=Alan+Hoffman+perfect+elimination+orderingampsource=blampots=UvM-wUMXF7ampsig=HsP83gL-ju07NlJfpiLQgJyrEXAamphl=trampsa=Xampei=WQNOVdrII8azUf7agMAMampved=0CCkQ6AEwAQv=onepageampq=Alan20Hoffman20perfect20elimination20orderingampf=false

Cordal Graph and Clique Trees

httpswwwmathbinghamtoneduzaslavOldcourses580S13blair-peytonchordal-graphs-clique-treesornl1992pdfcordal and their clique graphshttpwwwliafajussieufr~habibDocumentsWGChordauxpdf

Here we re go ing to v i sua l i ze m in ima l ve r tex sepa ra to r s

2 4

DEMO

REFERANCES

For MCShttppgmstanfordeduAlgspage-312pdfhttpwwwiiuibno~pinarMCS-Mpdfhttpwwwcsupcedu~valientegraph-00-01-epdf

For Perfect Elimination Orderingshttpwwwcisuoguelphca~sawadapaperschordalpdfhttpswwwmathbinghamtoneduzaslavOldcourses580S13bartlettMC2011_perfectgraphs_wk1_day3pdf

httpsbooksgooglefrbooksid=8bjSBQAAQBAJamppg=PA87amplpg=PA87ampdq=Alan+Hoffman+perfect+elimination+orderingampsource=blampots=UvM-wUMXF7ampsig=HsP83gL-ju07NlJfpiLQgJyrEXAamphl=trampsa=Xampei=WQNOVdrII8azUf7agMAMampved=0CCkQ6AEwAQv=onepageampq=Alan20Hoffman20perfect20elimination20orderingampf=false

Cordal Graph and Clique Trees

httpswwwmathbinghamtoneduzaslavOldcourses580S13blair-peytonchordal-graphs-clique-treesornl1992pdfcordal and their clique graphshttpwwwliafajussieufr~habibDocumentsWGChordauxpdf

REFERANCES

For MCShttppgmstanfordeduAlgspage-312pdfhttpwwwiiuibno~pinarMCS-Mpdfhttpwwwcsupcedu~valientegraph-00-01-epdf

For Perfect Elimination Orderingshttpwwwcisuoguelphca~sawadapaperschordalpdfhttpswwwmathbinghamtoneduzaslavOldcourses580S13bartlettMC2011_perfectgraphs_wk1_day3pdf

httpsbooksgooglefrbooksid=8bjSBQAAQBAJamppg=PA87amplpg=PA87ampdq=Alan+Hoffman+perfect+elimination+orderingampsource=blampots=UvM-wUMXF7ampsig=HsP83gL-ju07NlJfpiLQgJyrEXAamphl=trampsa=Xampei=WQNOVdrII8azUf7agMAMampved=0CCkQ6AEwAQv=onepageampq=Alan20Hoffman20perfect20elimination20orderingampf=false

Cordal Graph and Clique Trees

httpswwwmathbinghamtoneduzaslavOldcourses580S13blair-peytonchordal-graphs-clique-treesornl1992pdfcordal and their clique graphshttpwwwliafajussieufr~habibDocumentsWGChordauxpdf

REFERANCES

Cordal Graphs Thesis from University of Singapore Properties of Chordal Graphs

Graphs httpwwwmathbinghamtoneduzaslav581F14course-notes-chapter1pdf

Vertex SeperatorhttpenwikipediaorgwikiVertex_separatorhttpwwwncbinlmnihgovpmcarticlesPMC3927507f

Minimal vertex seperatorhttpwwwiiuibno~pinarMinTriSurveypdf

httpsbooksgooglefrbooksid=FKeoCAAAQBAJamppg=PA355amplpg=PA355ampdq=a+parra+and+p+scheffler+how+to+use+minimal+separators+of+a+chordal+graph+for+its+chordal+triangulationampsource=blampots=oKtjp-kRC8ampsig=j0o_Ir1_cOeAPrAp1glYf2DFy_Eamphl=trampsa=Xampei=SDlPVe2SOIaAUa3sgNABampved=0CDQQ6AEwAgv=onepageampq=a20parra20and20p20scheffler20how20to20use20minimal20separators20of20a20chordal20graph20for20its20chordal20triangulationampf=false

httpwwwmateunlpeduar~pdecaria57185pdf not useful

Generating all minimal vertex seperatorhttpwwwisimafr~berrygeneratingps

Minimal Stable Vertex Seperatorhttparxivorgpdf11032913pdf

Minimal Seperator at Cordal Graphhttppeopleidsiach~grandoniPubblicazioniCG06dmpdf

httpacels-cdncomS0166218X980012311-s20-S0166218X98001231-mainpdf_tid=51b34232-f708-11e4-86a2-00000aab0f27ampacdnat=1431257763_981ba2c5eac5bced0f5524a9b8e1a8f9

httpwwwsciencedirectcomsciencearticlepiiS0166218X10001824

REFERANCES

A linear time algorithm to list the minimal separators of chordal graphs1113147

L Sunil Chandran1 Fabrizio Grandoni httpwwwsciencedirectcomsciencearticlepiiS0166218X10001824

Important Source includes LEXBFS

Lexicographic Breadth First for Chordal Graphs Klaus Simon

httpciteseerxistpsueduviewdocdownloaddoi=1011565018amprep=rep1amptype=ps

httpwwwisimafr~berryRR-10-04

httpwwwcrraoaimscsorgresearchreportDownloads2014RR2014-15pdf

httpsbooksgooglefrbooksid=GbsPBwAAQBAJamppg=PA225amplpg=PA225ampdq=minimal+vertex+separator+lexbfsampsource=blampot

s=gK2RH1krfcampsig=pt-VraENVg3F1cxsCQcEwtsg7k8amphl=trampsa=Xampei=LUZPVY7ZCMLpUuv1gKAEampved=0CE0Q6AEwBQv=onepa

geampq=minimal20vertex20separator20lexbfsampf=false

httpsbooksgooglefrbooks

id=oDrTFgWtLdgCamppg=PA141amplpg=PA141ampdq=confluence+point+of+C+graphampsource=blampots=rMlrlenJuYampsig=4lVjpcXoPyQiVz

_rjKSgx5w_rcwamphl=trampsa=Xampei=WG1PVYLJOcv9UMLagOgMampved=0CCIQ6AEwAAv=onepageampq=confluence20point20of

20C20graphampf=false

httpistanfordedupubcstrreportscstr75531CS-TR-75-531pdf

httpwww2iiuibno~pinarMCSM-rpdf

httpwwwicsuciedu~agelfandpubPost-ProcessingtoReduceInducedWidthpdf

PEO Structure

httpwwwcsupcedu~valientegraph-00-01-epdf

httpwwwcseiitdernetin~naveencoursesCSL851uwaterloopdf

httpwwwiiuibno~pinarchordalpdf

ndash N A ZL I TEM UR

ldquoThank you for listeningrdquo

• Minimal Vertex Seperators amp All PEOs
• Cordal Graph Definition 2Perfect Elimination Ordering De
• CORDAL GRAPHS
• PERFECT ELIMINATION ORDERINGS
• VERTEX SEPERATORS
• VERTEX SEPERATORS (2)
• DETermining Cordality
• DETermining Cordality (2)
• Relation PEO-MVS
• NEw Proof of LEXBFS via MVs
• Slide 11
• HOW to GENERATE MVS
• KLB - LGB algorithms
• LEXbFS
• MOPLEXES in CHORDAL GRAPHS
• MOPLEXES in CHORDAL GRAPHS (2)
• Generating MVS via LEXBFS
• Generating MVS via LEXBFS (2)
• Generating MVS via McS
• HOW to GENERATE ALL PEOs
• Generating ALLPEOs VIA HAMILTON CL+PEO
• Generating MVS VIA HAMILTON CL+PEO
• Generating MVS VIA HAMILTON CL+PEO (2)
• Slide 24
• Slide 25
• Slide 26
• Slide 27
• Slide 28
• DEMONSTRATIONS
• Here we re going to visualize vertex separators 234 24
• Here we re going to visualize minimal vertex separators 24
• referances
• referances (2)
• referances (3)
• Slide 35