Bounded Combinatorial Width and Forbidden Substructures
by
Michael John Dinneen
B S University of Idaho
M S University of Victoria
A Dissertation Submitted in Partial Ful llment
of the Requirements for the Degree of
DOCTOR OF PHILOSOPHY
in the Department of Computer Science
We accept this dissertation as conforming
to the required standard
Dr Michael R Fellows Sup ervisor Department of Computer Science
Dr Hausi A Muller Department Memb er Department of Computer Science
Dr Jon C Muzio DepartmentMemb er Department of Computer Science
Dr Gary MacGillivray Outside Memb er Department of Math and Stats
Dr Arvind Gupta External Examiner Scho ol of Computing Science Simon Fraser
University
c
MICHAEL JOHN DINNEEN
University of Victoria
All rights reserved This dissertation may not b e repro duced in whole or in part
by mimeograph or other means without the p ermission of the author
ii
Sup ervisor M R Fellows
Abstract
A substantial part of the history of graph theory deals with the study and classi
cation of sets of graphs that share common prop erties One predominant trend is to
characterize graph families by sets of minimal forbidden graphs within some partial
ordering on the graphs For example the famous Kuratowski Theorem classi es
the planar graph family bytwo forbidden graphs in the top ological partial order
Most if not all of the current approaches for nding these forbidden substructure
characterizations use extensive and sp ecialized case analysis Thus until now for a
xed graph family thistyp e of mathematical theorem proving often required months
or even years of human e ort The main fo cus of this dissertation is to develop a
practical theory for automating with distributed computer programming this clas
sic part of graph theory We extend and more imp ortantly implementa variation
of the seminal work done byFellows and Langston regarding computing nite basis
characterizations
The recently celebrated Rob ertson Seymour Graph Minor Theorem establishes that
many natural graph families are characterizable bya nite set of graphs In particular
if a graph family is closed under the three basic minor op erations i e isolated vertex
deletions edge deletions and edge contractions then there exists by a nonconstruc
tive argument a nite set of forbidden graphs Two examples are the well known
k Vertex Cover and k Feedback Vertex Set graph families In this disserta
tion wecharacterize for the rst time these parameterized families among others
for small k
Our forbidden graph computations use a restricted search space consisting of graphs
of b ounded combinatorial width where pathwidth and treewidth are two imp ortant
metrics Using an algebraic enumeration sc heme for graphs wehave implementeda
terminating algorithm that wil l nd all minor order forbidden graphs for each xed
pathwidth For a targeted graph family this algorithm requires a mathematical
description given in one of many acceptable forms or combinations thereof suchasa
nite index congruence or a set of automaton generating tests Our main assumption
is that an upp er b ound on the pathwidth or treewidth of the largest forbidden graph
of a particular graph family is more readily available than its order or size
iii
Abypro duct of our b ounded width approachisthatwe give practical linear time
memb ership algorithms in the form of dynamic programs over parsed graph struc
tures of b ounded width for several graph families e g k Maximum Path Length
and Outer Planar
Examiners
Dr Michael R Fellows Sup ervisor Department of Computer Science
Dr Hausi A Muller Department Memb er Department of Computer Science
Dr Jon C Muzio DepartmentMemb er Department of Computer Science
Dr Gary MacGillivray Outside Memb er Department of Math and Stats
Dr Arvind Gupta External Examiner Scho ol of Computing Science Simon Fraser
University
iv
Contents
Abstract ii
Table of Contents iv
List of Figures viii
List of Tables xi
Acknowledgements xii
Intro duction
The Graph Theory Setting
Some Finitely Characterizable Graph Families
A Historical Persp ective on Computing Obstructions
An Overview of Our Computational Technique
A Survey of The Dissertation
I The Basic Theory
Bounded Combinatorial Width
Intro duction
Treewidth and k trees
Pathwidth and k paths
Other graph theoretical widths
Algebraic Graph Representations
v
The t parse op erator set
Some t parse examples
Other op erator sets
Simple Enumeration Schemes
Canonic pathwidth t parse s
Canonic treewidth t parses
Graph Minors and Well Quasi Orders
Preliminaries
The Graph Minor Theorem
Other Graph Partial Orders
Finding Forbidden Minors
Key t parse Prop erties
A Simple Pro cedure for Finding Obstructions
Proving t parses Minimal or Nonminimal
Direct pro ofs of nonminimality
Pro ofs based on a dynamic programming algorithm
Pro ofs obtained by a randomized search
Pro ofs based on a testset congruence
Making the Theory Practical
Pruning at disconnected t parses
Searching via universal distinguishers
Using other nite index congruences
Finding uses of randomization
The Implementation and The Future
Using Distributed Programming
Software Summary
Future Research Goals
Second order congruence research
Approximation algorithms based on partial obstruction sets
vi
II Obstruction Set Characterizations
Vertex Cover VC
Intro duction
A Finite State Algorithm
The VC Obstruction Set Computation
The VC Obstructions
Indep endent Set and Clique Families
FeedbackVertex Edge Sets FVS and FES
Intro duction
The FVS Obstruction Set Computation
A nite state algorithm
A complete FVS testset
The FVS Obstructions
The FES Obstruction Set Computation
A direct nonminimal FES test
A complete FES testset
The FES Obstructions
Some Generalized VC and FVS Graph Families
Path Covers
A path cover congruence
Cycle Covers
Path Cycle Cover Testsets
Some maximum path cycle testsets
A maximum path automaton example
Some generic cycle cover testsets
Other VC FVS Generalizations
The Path and Cycle Cover Obstructions
Outer Planar Graphs
Intro duction
vii
The Outer Planar Computation
An outer planar congruence
A nite state algorithm
The Outer Planar Obstructions
Some other surface obstructions
III Applied Connections
Pathwidth and Biology
VLSI Layouts and DNA Physical Mappings
An Equivalence Pro of
Some Comments
Automata and Testsets
Intro duction
Finding a Minimum Testset is NP hard
A Testset Example Building Memb ership Automata
Using tree automata for b ounded treewidth
Quickly Finding Obstructions using Automata
Computing Pathwidth byPebbling
Preliminaries
A Simple Linear Time Pathwidth Algorithm
Further Directions
Conclusion
A Summary of the Main Results
Two Key Applications
Annotated Bibliography
Index
viii
List of Figures
Kuratowski s planar obstructions
Illustrating a partial order of graphs and a family s obstructions O F
Pro jective plane emb eddings of Kuratowski s planar obstructions
Induced forbidden chordal graphs asteroidal triples
Demonstrating the edge contraction op eration for the minor order
All connected minor order obstructions for k Edge Bounded IndSet
for k
Some non trivial knots
The minor order obstructions for the linkless graph family
Two graphs that are memb ers of b oth the Vertex Cover and
Feedback Vertex Set graph families
Example of our enumeration order for graphs of pathwidth
Actual search tree for the Edge Bounded IndSet graph family s
obstructions pathwidth
Demonstrating graphs with b ounded treewidth and pathwidth
A partial order of several b ounded width families of graphs
The obstructions to a pathwidth and b treewidth
The minor order obstructions to treewidth
Illustrating an edge contraction p oset
A map from b oundaried graphs to edge colored graphs
Illustrating the a lift and b fracture graph op erations
The weak immersion order obstruction set for Cutwidth
Atypical t parse search tree each edge denotes one op erator
ix
A general indep endence algorithm I d p for pathwidth t parses
The set T of b oundaried graph tests for Hamiltonicity
HC
Schematic view of our distributed obstruction set software
Some available run time options for obstruction set computations
Building an NFA that accepts the union of minor containment families
A general vertex cover algorithm for t parses
Actual search tree for Vertex Cover with graphs of pathwidth
Connected obstructions for and VertexCover
Connected obstructions for Vertex Cover
Connected obstructions for Vertex Cover
Connected obstructions for Vertex Cover
A general feedbackvertex set algorithm for t parses
Connected obstructions for Feedback Edge Set
Connected obstructions for Feedback Vertex Set
Connected obstructions for Feedback Edge Set
Connected obstructions for Feedback Vertex Set pathwidth
Known connected obstructions for Feedback Edge Set
Biconnected Feedback Edge Set obstructions without degree
vertices pathwidth
Biconnected Feedback Edge Set obstructions without degree
vertices pathwidth
The vertex op erator sub cases for the pro of of Theorem
Illustrating the p case for the pro of of Theorem
i
The smallest biconnected non Hamiltonian graph
Four typ es of b oundaried tests for the MaxPath p and MaxCycle l
graph families l p
corresp onding to Table The automaton for MaxPath
k
A t b oundaried test example in fork Cycle Cover
t
Connected obstructions for Path Cover
Connected obstructions for Path Cover
x
Known connected obstructions for Path Cover
Known connected obstructions for Path Cover
Connected obstructions for Cycle Cover pathwidth
All vertex outer planar dual trees duals minus outer face vertex
The smallest outer planar graph with pathwidth
Illustrating the pro of of Lemma
A commutative diagram for the pro of of Theorem
Known connected obstructions for Outer Planar
Figure continued Outer Planar obstructions
All connected obstructions with at most vertices for Planar
Figure continued small obstructions for Planar
A dense symmetric toroidal obstruction with pathwidth
The three smallest vertices toroidal obstructions
Illustrating the k CVS and k ICG problems
The AMT automaton M built from an MTC instance
A graph connectivity testset for b oundaried graphs
A graph connectivitymemb ership automaton for parses
Emb edding pathwidth tree obstructions Tree t in binary trees
Illustrating our linear time pathwidth algorithm
xi
List of Tables
Numb er of connected and disconnected minor order obstructions for
k Edge Bounded IndSet fork
Source co de breakdown by area for our VACS software sub directories
Vertex cover state tables computed columns for Example
Summary of obstruction set computation for vertex cover
Feedbackvertex set state tables computed for Example
Summary of our Feedback Vertex Set obstruction set computa
tion pathwidth
The transition diagram for the MaxPath automaton
The M er g eGaps function for the outer planar algorithm
Tractability b ounds for various parameterized lower ideals
xii
Acknowledgements
Iacknowledge my thesis advisor Dr MikeFellows for intro ducing the topic
for this dissertation and b eing enthusiastic ab out the results Sp ecial thanks
are due to my friend Dr Kevin Cattell who contributed several years of C
programming and researchtowards the realization of our software pro ject
Without him many of the following results would still b e on the waiting list
Ithankmy Los Alamos mentor Dr Vance Fab er for providing me with an
opp ortunity to do research in a pro ductive atmosphere including access to
the vast computer resources at The Laboratory I thank Dr Arvind Gupta
for b eing my external examiner and for his detailed comments I also thank all
of my friends at the University of Victoria notably Todd Wareham and in the
United States for the relaxing times away from research Lastly I appreciate
my family in Idaho for enduring my transient lifestyle
Kevin Cattell assisted me in the following areas
He programmed all of the database co de the t parse canonicityal
gorithm some of the primitive t parse algorithms e g our edge
contraction algorithm on page and several shell scripts
He contributed valuable researchtowards our vertex cover characteriza
tions CD which includes some of the results of Chapter vigorously
pro of read our CDF a pap er and help ed prove that our pathwidth al
gorithm runs in linear time CDF b
He monitored jointly several obstruction set searches and often ne
tuned the software for b etter e ciency Some of our searches required
many time consuming restarts and to ok several months of CPU time to complete
Chapter
Intro duction
One of the most famous results in graph theory is Kuratowski s characterization of
planar graphs a graph is planar if and only if it contains neither the complete
bipartite graph K nor the complete graph K Theobstruction set for planarity
thus consists of these two graphs which are displayed b elow in Figure These
graphs are almost planar or the smallest non planar
K
K
Figure Kuratowski s planar obstructions
This example indicates the form of all obstruction set characterizations of graph
families for some xed graph family denoted by F a graph G is a member of F if
and only if G do es not contain as a substructure anymemb er of some set of graphs
O F fO O g The precise meaning of substructure mayvary For these
typ es of characterizations graphs are related by some partial order as illustrated
in Figure by a Hasse diagram where each no de representing some graph is
structurally contained in the ones connected ab ove it Here any non family graph
CHAPTER INTRODUCTION
F
F
obstructions
F
graph family
smallest graph or K
Figure Illustrating a partial order of graphs and a family s obstructions O F
ab ovethelower dashed line i e any graph represented by a black or gray no de
contains at least one obstruction black no de Notice that by de nition these
typ es of orderings of graphs are transitiveintheupward direction even though all
relationships are not shown in the gure Each edge in Figure can b e viewed as
one primitive graph reduction b etween a bigger graph and a smaller graph Di erent
combinations of these reductions on a given graph may yield the same graph e g
note that edges can cross over the obstruction set b oundary
Classifying graphs by forbidden substructures has b een an imp ortant part of
research in graph theory ever since its inception and there are manycharacterization
theorems of this kind Perhaps a familiar example is the set of acyclic graphs i e
those graphs without a self connecting path of edges Here the forbidden subgraphs
are the individual cycles of various lengths fC j i g Ifwe add more structure
i to this subgraph partial order then there is just one forbidden substructure namely
CHAPTER INTRODUCTION
the smallest cycle C This is accomplished by adding sub divided edge relationships
on graphs i e a graph with a vertex inserted on some edge is structurally ab ovethe
original graph Wenow presenttwo famous examples based on two di erent partial
orders These orders are formally de ned later in Chapter
The rst graph emb edding analog to Kuratowski s characterization of planar
graphs was presented in GHW b for the real pro jective plane M obius band
lid This simplest surface after the plane sphere can b e viewed as a plane with
a circular b oundary where diagonally opp osite b oundary p oints are identi ed e g
see FG Sti Below in Figure we easily show graph emb eddings without
edge crossings of b oth the planar obstructions K and K this indicates that the
family of pro jective planar graphs is larger than the family of planar graphs For this
non orientable surface an obstruction set of irreducible graphs of GHW a in
the top ological homeomorphic partial order was proven complete in Arc From
these obstructions there are minor order forbidden graphs Arc Mah
b
a b a
c
c
a b a b
K
K
Figure Pro jective plane emb eddings of Kuratowski s planar obstructions
Our next example is drawn from computational biology We consider a graph
to b e chordal if all induced cycles of length four or greater contain an internal cross
edge chord An indep endent triple of vertices is called an asteroidal triple if b etween
every pair in the triple there exists a path that avoids the neighb orho o d of the third
The four forbidden asteroidal triples see Figure and the simple cycle C prevent
a graph from b eing represented as an intersection of interval line segments Weg
Fel This nite classifying set of minimal graphs is obtained from the general
CHAPTER INTRODUCTION
Figure Induced forbidden chordal graphs asteroidal triples
in nite family of asteroidal triples this earlier result based on the v ertex induced
subgraph partial order showed that interval graphs are characterized as graphs that
are chordal and free of asteroidal triples LB Mir
Several other examples of classifying graphs by forbidden substructures can b e
found in APC CKK Fis LPS MS Pro RS c Most of the known
obstruction set characterizations were di cult to prove and proven solely by classical
pap er and p encil metho ds e g see KL GH This situation is nowchanging as
computers are b eing incorp orated into the research pro cess For example the elusive
target of nding the obstruction set for the family of torus emb eddable doughnut s
surface emb eddable graphs now app ears to b e closer by the new theories of automated
mathematical theorem proving
This dissertation is motivated by the recent results of Rob ertson and Seymour
RS a on the well partial ordering of graphs under the minor and other orders
They have non constructively established that many natural graph prop erties have
Kuratowski typ e characterizations That is these graph families can b e character
ized by nite obstruction sets in an appropriate partial order The rst few pap ers of
their comprehensive graph minors series may b e found in RS RSf Our work has
fo cused on nding obstruction sets for these typ es of characterizable graph families
The following sections survey our computational approach for nding obstruction
sets We b egin with a review of the minor partial order for graphs Then in Section
weintro duce two kinds of k parameterized graph families that form the basis of many
of our graph family characterizations Next in Sections and we explain howto
e ciently search for obstruction sets within a b ounded combinatorial width domain
At the end of this intro duction we outline the remaining chapters of this dissertation
CHAPTER INTRODUCTION
The Graph Theory Setting
The reader is assumed to have a familiarity with graph theory asmay b e obtained
from any one of these intro ductory graph theory texts BM CL HR Tru
Wil Wenowintro duce a few of our basic graph theoretical conventions
We are primarily interested in nite simple undirected graphs i e those nite
graphs without lo ops and multi edges A graph G V E in our context consists
of a nite set of vertices V and a set of edges E where each edge is an unordered
pair of vertices We use the variables n sometimes jGj andm to denote the order
number of vertices and size numb er of edges of a graph A graph is connected
if there exists a path b etween every twovertices A tree is a connected graph with
m n that is it contains no cycles but has a unique path b etween every two
vertices Other formal graph theoretic de nitions are presented later as needed
This dissertation fo cuses on a partial order based on graph minors Sp eci cally
a graph H is a minor of a graph G denoted H G ifH can b e obtained by
m
contracting some p ossibly zero edges in a subgraph of G Figure illustrates the
edge contraction op eration Recall that any partial order of graphs including the
minor order satis es three key prop erties It is re exive G G transitive if
m
G G and G G then G G and anti symmetric if G G and G G
then G G
The most imp ortant application of the celebrated Graph Minor Theorem GMT
by Rob ertson and Seymour formerly known as Wagner s Conjecture is that many
graph families are easily shown to have p olynomial time memb ership algorithms
Here the GMT guarantees that many graph families have only a nite number of
obstructions These memb ership algorithms are based on an O n algorithm that
can check if a xed graph is a minor of another graph RS b Hence in theory to
check for memb ership in an applicable graph family F one just runs this p olynomial
time minor checking algorithm once for each obstruction in O F An input graph G
is a member of F if and only if G has none of these obstructions as a minor Thus
once the promised nite set of obstructions O F is found for F a constructive
memb ership algorithm exists However it is still op en whether these new algorithms
can b e made practical This is b ecause there are astronomically large hidden constants
CHAPTER INTRODUCTION
First collapse edge while preserving adjacencies
Then removeanyincidentmulti edges that o ccur
K
m
K
m
Figure Demonstrating the edge contraction op eration for the minor order
in this cubic time minor containment complexity result FRS Also the number of
obstructions may b e prohibitively large
To indicate what typ es of graphs interest us weinvestigate a contrived but non
trivial graph family Recall that the independence of a graph is the maximum number
of mutually non adjacentvertices i e the cardinality of the largest indep endent
set It is easy to show that any minor of a graph with indep endence size k
also satis es this inequality For example after deleting an edge from a graph the
indep endence can increase by at most one implying that the sum will not increase
For any xed k let us call this graph family k Edge Bounded IndSet If one do es
not observe that this family has a nite number of memb ers recognizing graphs in
k m
k Edge Bounded IndSet may seem to require a running time of O n That
is where the size m has b een previously computed an algorithm could checktoseeif
ertex subsets of cardinality k m is an indep endent set An application any of the v
CHAPTER INTRODUCTION
k k k k
k k
Figure All connected minor order obstructions for k Edge Bounded IndSet
for k
of the GMT improves this brute force algorithm to O n time based on checking a
nite set of forbidden minors
For each small xed k an obstruction set for k Edge Bounded IndSet is easily
found by the following logic Weknowthat k isolated vertices is an obstruction
so k also b ounds the order of the largest obstruction We display all the connected
obstructions for Edge Bounded IndSet through Edge Bounded IndSet in
Figure One of our general results of Chapter shows how disconnected obstruc
tions are obtained For example the union P K of a path of length and an
isolated vertex is an obstruction for Edge Bounded IndSet
We can sometimes exploit other prop erties of graph families to reduce the ab ove
mentioned p olynomial time complexity We use the p opular notions of treewidth and
pathwidth also intro duced by Rob ertson and Seymour RS RS c to combinato
rially b ound certain subsets of a graph family Informally graphs of width at most k
are subgraphs of tree like or path like graphs where the vertices form cliques of size
at most k These combinatorial widths are discussed in detail in Chapter
When restricted to graphs of treewidth or pathwidth at most k there exists a linear
CHAPTER INTRODUCTION
Figure Some non trivial knots
time algorithm to check if a xed graph is a minor of the input ALS That is
these algorithms run in time O f k n where f is some computable function A
well noted example recently improved by Bo dlaender is that all nitely characteri
zable minor order graph families that exclude at least one planar graph have linear
time memb ership algorithms Bo d a That is for a xed planar graph P either a
graph has P as a minor or has treewidth at most some function of jP j RS a
For a concrete example of a guaranteed linear time memb ership algorithm con
sider our k Edge Bounded IndSet families We see that all of these families have
a planar obstruction e g the planar graph consisting of k isolated vertices
Thus by Bo dlaender s result there exist linear time recognition algorithms for these
graph families Also known is a linear time indep endent set algorithm for graphs of
b ounded treewidth e g see Bo d c
As mentioned earlier a constanttimememb ership algorithm is known for each
k Edge Bounded IndSet graph family since each has a nite function of k num
b er of graph memb ers In particular if more than k edges are read for an input
graph G then weknowthat G is not a member of k Edge Bounded IndSet So
the yes instance input must b e of constant size
A remarkable consequence of the GMT is that for some graph families that were
not even known to b e decidable the GMT can b e applied to prove the existence of
a p olynomial time recognition algorithm see F el foranicesurvey For example
consider the set of graphs that can b e emb edded in dimensional space such that ev
ery cycle is not knotted in the everyday sense as depicted by the knots in Figure
It is not hard to see that this family of knotless graphs whichcontains the planar
graph family is closed under minors i e taking a subgraph or contracting an edge
CHAPTER INTRODUCTION
Figure The minor order obstructions for the linkless graph family
can not create a knot with resp ect to a knotless emb edding The complete graph
K is an example of a graph that contains a least one nontrivial knot no matter how
it is emb edded in space CG However it is not known if the partite graph
K has a knotless emb edding Tho The promised nite obstruction set for
this family is still unknown However a related knot theory family of graphs those
graphs with a linkless spaceembedding has recently b een characterized by the seven
forbidden minors shown in Figure MRS RST a RST b RST c
Some Finitely Characterizable Graph Families
As a consequence of the Graph Minor Theorem any family of graphs F that is closed
under minors i e if H G and G F then H F has an obstruction set O F
m
of nite cardinality More generally a family of graphs F that is closed relativetoa
partial order is called a lower ideal
Likethe k Edge Bounded IndSet families many minor order lower ideals are
parameterized byaninteger k The previously mentioned planar and toroidal graph
families are surface emb eddable families for xed genus i e the numb er of sphere
handles k and k resp ectively CL Tru In fact our principle graph
families e g k Vertex Cover and k Feedback Vertex Set for which our
computational technique is illustrated fall into this framework A k parameterized
CHAPTER INTRODUCTION
VC f g VC f g or f g
FVS f g f g or f g FVS f g f g f g
f g f g or f g
Figure Two graphs that are memb ers of b oth the Vertex Cover and
Feedba ck Vertex Set graph families
lower ideal F is de ned by some typeofgraphinvariant function that maps graphs
to integers such that whenever H G wealsohave H G In this general
m
framework a graph family F equals fG j G k g for some integer constant k
Figure illustrates twomemb ers of b oth the graph families Vertex Cover
and Feedback Vertex Set by displaying eachminimum vertex cover VC and
feedbackvertex set FVS The family Vertex Cover is the set of graphs that
havea VC of cardinality at most i e every edge is incident to at least one of the
three cover vertices VC Likewise a graph G is in Feedback Vertex Set if
there exists an FVS with twovertices u and v suchthatG nfu v g is acyclic i e this
subgraph contains no cycles
Given a minor order lower ideal F often but not always another class fk
Fjk g of parameterized and nitely characterizable graph families is obtained
by de ning lower ideals that are within k vertices or edges of F That is for a xed
family F the following parameterized families are lower ideals e g see FL b for
pro of
k F fG V E j G n V F where V V and jV j k g
v
And the following families mayalsobelower ideals
k F fG V E j G n E F where E E and jE j k g e
CHAPTER INTRODUCTION
Table Numb er of connected and disconnected minor order obstructions for
k Edge Bounded IndSet for k
k
connected
disconnected
total obstructions
For example in Chapter we study the lower ideal Outer Planar that is based
on those graphs that are within one vertex of the family of outer planar graphs
Unfortunately the corresp onding family of graphs that are at most one edge away
from b eing outer planar is not a minor order lower ideal In Chapter welookat
the lower ideals k Feedback Edge Set that are those graphs that are within k
edges of acyclic i e each graph in this family has a set of at most k edges when
removed pro duces a forest
There is a tendency for the numb er of obstructions for natural parameterized
families to grow explosively as a function of the parameter k For example the num
b er of minor order obstructions for k Pathwidth i e graphs with pathwidth at
most k is for k for k and provably more than million for k
KL Even lo oking at our small k Edge Bounded IndSet graph families the
counts given in Table indicate exp onential growth in the numb er of obstructions
One goal of our research goal is to explore the following working hyp othesis
Natural forbidden substructure theorems of feasible size are feasibly computable
We susp ect that the toroidal obstruction set which has a pro jected size of ab out
graphs is feasibly computable by the metho ds of this dissertation
Of course if a lower ideal is not parameterized the feasibility of computing
its obstruction set will have to b e based on other criteria The family of knotless
graphs is one suchinteresting example where we need an alternate metho d to predict
the size of the obstruction set Again in any of these non parameterized cases if the obstruction set is small and in conjunction with a few other family sp eci c
CHAPTER INTRODUCTION
ingredients we conjecture that it should b e feasibly computable
A Historical Persp ective on Computing
Obstructions
Most of the previous techniques for computing obstructions have b een sp eci c to a
particular lower ideal In particular several Ph D dissertations fo cus on characteriz
ing just one graph family and use strictly mathematical case analysis see for example
Arc Kin Sar For some of these characterizations computers are used for
the most part to mainly aid in the veri cation pro cess usually byweeding through
a manageable known set of p otentially minimal forbidden graphs
The basic results for our general purp ose computational approach originated from
the foundational pap er byFellows and Langston FL a which used the Graph
Minor Theorem to prove termination of a nite state search pro cedure This approach
for computing obstruction sets requires additional problem sp eci c results These
results are nontrivial but seem to b e generally available in one form or another for
virtually every natural lower ideal F TheFellows Langston approach for computing
obstructions requires the following three ingredients
i a decision algorithm for F
ii a nite index congruence that re nes the canonical congruence for F and
iii a b ound on the maximum treewidth of the obstructions for F
The ab ove mentioned regular language canonical congruence is intro duced in the
next section
In general we knowthathaving only the rst ingredient is not su cientto
compute the obstruction set for an arbitrary lower ideal F e g see vL FL b
That is from only a decision algorithm for F there is no algorithm to compute O F
We conjecture that ha ving the rst two ingredients may b e su cient CDDF This
dissertation uses these three basic ingredients but in a more feasible way than the
can b e computed results of FL a
Wenowmention two other successful e orts concerning the theory of computing
obstructions Lagergren and Arnb org in LA constructively show i e without
CHAPTER INTRODUCTION
the GMT that every b ounded treewidth minor order lower ideal with a nite index
family congruence has a nite numb er of obstructions Also for b ounded treewidth
Courcelle has indirectly proved that the obstructions for anylower ideal de ned bya
second order monadic logic formula can b e computed he sho wed a metho d of pro duc
ing the ab ove mentioned nite index congruence from this formula Cou a Cou b
We do not b elieve that either of these theoretical metho ds has b een explored in the
implementation sense
One of the goals for this dissertation is to bridge the gap b etween the theory of
obstruction set computations and their practical implementations
An Overview of Our Computational Technique
This dissertation describ es a basic theory for computing obstruction sets whichis
partially automatable for any minor order lower ideal Wehave successfully used our
technique to nd obstruction sets for several typ es of graph families and these are
presented in Part I I of this dissertation
Wenowsketch our metho d for computing obstruction sets To nd minor order
obstructions it is su cient to nd the obstructions for a slightly relaxed partial order
called the boundary minor order for graphs with a constantnumb er of lab eled ver
tices denoting the b oundary It is easy to extract ordinary obstructions from the
b oundaried obstructions
Wehaveanenumeration scheme that generates all of the b oundaried graphs for
a sp eci ed maximum pathwidth or treewidth Our tree likeenumeration scheme
has an imp ortant prop ertythatallows us to prune at dead ends during the search
pro cess W ehave a pre x prop erty on b oundaried graphs with resp ect to our
enumeration order that ensures that it is safe to prune the search space this way
Thus only fruitful branches to b oundaried obstructions are taken thereby reducing
the overall computation time Our enumeration tree is obtained from a partial order
of b oundaried graphs This order is di erent than the substructure partial order
in this case the b oundary minor order that was illustrated earlier in Figure
Figure shows the b eginning of an enumeration tree and partial order for
CHAPTER INTRODUCTION
Continues upward
a canonic search tree
redundant
b oundaried
graphs
redundant paths
Figure Example of our enumeration order for graphs of pathwidth
graphs of pathwidth Within this order we grow the searchtreeby following the
canonic paths upward Also a graph may b e assigned a redundant lab el if every
obstruction ab ove that graph is obtainable by expanding out from another graph
This initial underlying search tree is indep endentofany sp eci c lower ideal that we
are trying to characterize
For our main pre x pruning pro cess we require one or more family sp eci c algo
rithms These easily obtainable requirements are used to determine graph minimal
ity Here with resp ect to a nite index congruence for a lower ideal F a b oundaried
graph G is minimal if no b oundaried minor H of G is in the same equivalence class
as G Our enumeration tree of b oundaried graphs has the pre x property that ev ery
parent of a minimal graph is also minimal
In order to de ne graph minimality we use an equivalence class on b oundaried
graphs of the same b oundary size with resp ect to a lower ideal F as follows First
we glue together two b oundaried graphs with the binary op erator by coalescing
b oundary vertices with the same lab el Then two graphs G and H are in the same
equivalence class of this canonical congruence if and only if for every b oundaried
CHAPTER INTRODUCTION
graph Z called an extension
G Z F H Z F
For the set of graphs of b ounded treewidth or pathwidth this canonical congruence
is nite index for any minor order lower ideal e g see AF
During our search any graph ab ove a nonminimal graph in the enumeration
order recall Figure is not generated Thus many graphs b oth in and out of
the targeted lower ideal are avoided altogether Because of our pre x prop ertyand
the GMT weknow that only a nite numb er of graphs will b e enumerated for each
b ounded width search Thus if the largest width of any obstruction is known for a
targeted lower ideal then we can nd the complete obstruction set For example we
know that the obstructions for the k Vertex Cover graph family have pathwidth
at most k
Fora avor of what our search trees lo oks like wenowgive a concrete example
The exact details of what follows may not b e totally clear at this juncture Our
enumeration tree for the Edge Bounded IndSet family is shown in Figure
The no de lab els are indices into a search database and are sequenced in enu
meration order The white no des b oth the circles in family and squares out of
family represent minimal graphs as de ned ab ove That is the v e white squares
denote the b oundaried minor order obstructions to Edge Bounded IndSet The
two minor order obstructions that where shown earlier in Figure are K
and C These can b e extracted by following the no de op erator lab els
up the tree The other three b oundaried obstructions represent the same discon
nected obstruction P K but with di erent b oundaries The ro ot represents
the smallest b oundaried graph consisting of three isolated b oundary vertices The
missing no de numb ers in this tree are either irrelevant e g disconnected or
non canonic redundant graphs The only nonminimal family graph P is
congruent to one of its edge deleted minors K P Lastly we p oint out that
graph the cycle C is an example of a minimal graph that do es not lead to
some b oundaried obstruction in the search tree
Next wesurvey our four metho ds for determining graph minimality with resp ect
to the canonical congruence for a lower ideal F A main contribution of this dis
CHAPTER INTRODUCTION
#52,01 #54,12 #61,0 #62,1 #65,02 #66,12 #56,1
#31,0 #32,1 #35,02 #36,12 #25,0 #44,1 #46,01 #48,12
#16,01 #17,02 #19,0 #24,12
#7,0 #11,02
#4,01
#0,[0,1,2]
Figure Actual search tree for the Edge Bounded IndSet graph family s
obstructions pathwidth
sertation is to combine these metho ds to make obstruction set computations more
feasible
Direct nonminimalitychecking After testing for some structural or graph invari
ant prop erty we sometimes know when a graph has a congruent minor with resp ect
to the canonical congruence Often these prop erties are quickly computed For ex
ample in the family k Edge Bounded IndSet we can show that a graph G with
an isolated K is nonminimal since the minor obtained by deleting one of its edges is
congruentto G
A nite state dynamic program Given an algebraic representation for a b ound
aried graph of b ounded width we use a nite state dynamic program to determine
whether a graph G and a minor G are congruent These algorithms are designed
so that whenever two input graphs fall in the same computation state they are con
CHAPTER INTRODUCTION
gruent By the nature of the dynamic program b oth G Z and G Z will fall
in the same state for any b oundaried graph Z Unfortunately in most cases if two
graphs fall into di erent states we can not infer that they are not congruent For
our k Edge Bounded IndSet example a congruence that can b e implementedas
a dynamic program is now presented For a b oundaried graph G and eachvertex
subset S of the b oundary de ne the following partial function
I ndS et S max jI j k I S and I is an indep endent set
G
Two b oundaried graphs G and H are in the same equivalence class in this re nement
of the canonical congruence for k Edge Bounded IndSet if
min jE G j k min jE H j k and
I ndS et S I ndS et S for all S
G H
We later present and prove correct a linear time dynamic program for the indep en
dent set comp onent of this congruence see Chapter
A random extension search A random b oundaried graph Z may b e a witness to
G not b eing congruent to a minor G That is we search for an extension Z such
that G Z is not in F while G Z is in F Very little computer resources are
needed for these typ es of checks We rep eat these random extension searches several
times until each minor G of G is shown to b e in a di erentequivalence class than
G s equivalence class or until a give up threshold is reached Since F is a lower
ideal it is b ene cial to nd a random b oundaried graph Z suchthat G Z is not
in F but for anyminorZ of Z G Z is in F This is done b efore checking if an
extended minor G Z is in F
Boundaried graph tests We use a nite subset of b oundaried graphs called
a testset that is su cient to determine if two graphs are in the same canonical
equivalence class That is if two graphs G and H pass the same set of tests via
then they are congruent for all extensions otherwise they are not A graph G passes
a test Z whenever G Z is in F The reader may recall the Myhill Nero de Theorem
regarding regular languages Usually a testset for a lower ideal F is quite large
Thus determining minimality or nonminimality for a particular graph may require
many family memb ership checks We can often reduce the cardinality of a testset
CHAPTER INTRODUCTION
by observing a few prop erties of F For the family k Edge Bounded IndSet we
knowthatany useful test requires at most k edges Furthermore for this lower
ideal any comp onent C of a test Z without any b oundary vertices can b e replaced
with I ndependence C jE C j isolated vertices This means that all edges of each
test for this family should b elong on some path to a b oundary vertex
Weusevarious selections of the ab ovementioned minimality nonminimality
metho ds in Part I I of this dissertation to characterize several interesting graph
families As alluded to earlier nding obstruction set characterizations is in princi
ple one route for establishing constructiveversions of many of the known applications
of the Rob ertson Seymour results
ASurvey of The Dissertation
The primary and secondary researchcontributions in this dissertation are
Finding obstruction set characterizations for various families of graphs using
newly develop ed computational techniques
Prove usable combinatorial width b ounds for lower ideals
Intro duce practical graph enumeration schemes
using algebraic representations of graphs
Use compact nite index family congruences
Pro vide canonical congruences in the form of testsets
Utilize a randomized obstruction set search strategy
Developing e cient algorithms for graphs of b ounded combinatorial width
Give dynamic programs for several graph families
Prove a simple linear time algorithm for nding path decomp ositions of
small width
Showhow family memb ership automata for graphs can b e constructed from b ounded width nite index congruences
CHAPTER INTRODUCTION
For the rst research area i e our more glamorous and develop ed area we
implemented an automated technique for nding obstruction sets In the next four
chapters we describ e some of our practical exp erience in realizing this goal e g
what new theory was needed In the four chapters that follow this general theory
we illustrate our approachbycharacterizing several graph families We also present
several required family sp eci c results that were needed for the completion of these
obstruction set computations
The second research area is more applied in nature It has b een known in several
research circles that e cient graph algorithms often exist when the input is restricted
to graphs of b ounded combinatorial width This is contrasted with the general case
where the same computational problems app ear to b e intractable Many automated
techniques based on formal problem descriptions exist in theoretical form only for
pro ducing these typ es of b ounded width algorithms e g see Bor Cou Un
til these approaches are develop ed we rely on the successful dynamic programming
metho ds e g see ALS Bo d a CSTV This dissertation contributes new
practical algorithms to this list That is for each of our obstruction set compu
tations wealsogive a corresp onding linear time memb ership algorithm for parsed
graphs of b ounded pathwidth
Asurvey of the remaining chapters of this three part dissertation is given b elow
At this time we p oint out that some of the material of this dissertation has ap
p eared elsewhere notably Chapters and in the pap ers CD CDF a and
CDF b resp ectively After a concluding Chapter an annotated bibliographyis
given for the sake of completeness An index is also included at the end
The rst part of this dissertation contains our basic theory for computing obstruc
tion sets based on a b ounded combinatorial width search space Chapter shows how
graphs can b e algebraically represented in this b ounded width domain Chapter
provides the set theoretic background needed for nding minor order obstructions
Chapter contains our main theory for nding obstructions with implementation
asp ects in mind Chapter concludes Part I with a brief description of our software
and a selection of future research topics
CHAPTER INTRODUCTION
The second part of this dissertation illustrates our computational technique for
nding obstruction sets This coverage spans several graph families We b egin in
Chapter bycharacterizing the well known families k Vertex Cover fork
Next in Chapter we illustrate the use of Myhill Nero de testsets con
sisting of b oundaried graph tests to compute obstruction sets for twotyp es of almost
tree families of graphs i e k Feedback Vertex Set and k Feedback Edge Set
In the third chapter in this series namely Chapter we generalize the lower ideals of
the previous twochapters by studying those graphs with small path covers and cycle
covers To round out our presentation wegive in Chapter some initial results on
computing obstructions for surface emb edable graph families with the main fo cus on
the family Outer Planar
The third part of this dissertation presents several applications oriented topics
concerning nite state automata and b ounded combinatorial width Each of these
chapters is indep endent from the remaining parts of this dissertation and can b e di
gested without a full understanding of our obstruction set computation theory Chap
ter contains a link b etween computational biology and VLSI layout problems in
terms of b ounded pathwidth Chapter develops some complexity results regarding
minimizing testsets for automata and shows how automata can b e used to compute
obstruction sets Lastly Chapter presents a very simple linear time algorithm for
nding small path decomp ositions of a graph or determines that a graph has large
pathwidth
Part I
The Basic Theory
Chapter
Bounded Combinatorial Width
Weareinterested in certain subsets of the set of all nite graphs In particular two
p opular graph families that we utilize have b ounded combinatorial width Here
the width of a graph is de ned in many p ossible ways often in uenced by the graph
problems that are b eing investigated For example for VLSI layout problems wemay
de ne the width of a planar graph which represents a circuit as the least surface
area needed over all layouts where vertices and edge connections are laid out on a
grid Two sets of graphs that we utilize in this dissertation are graphs of b ounded
pathwidth and treewidth These width metrics and the corresp onding graph families
that are b ounded by some xed width are imp ortantfora variety of reasons
These families are minor order lower ideals
They play a fundamental role in the pro of of the Graph Minor Theorem
Many p opular classes of graphs are subsets of these families
These widths are used to cop e with intractability
There are many natural applications
The rst statement is particularly imp ortant for our obstruction set computations
Here if H is a minor of a graph G then the pathwidth treewidth of H is at most
the pathwidth treewidth of G
CHAPTER BOUNDED COMBINATORIAL WIDTH
Regarding the development of graph algorithms wewant to emphasize that there
is a common trend in that practical algorithms often exist for restricted families of
graphs such as trees or planar graphs while the general problems remain hard to
solve for arbitrary input This tendency for e cient algorithms includes instances
where the problem s input is restricted to graphs of b ounded pathwidth or b ounded
treewidth
Intro duction
The use of b ounded combinatorial width is imp ortant in suchdiverse areas as com
putational biology linguistics and VLSI layout design KS KT M oh For
problems in these areas the graph input often has a sp ecial tree like or path like
structure A measure of width of these structures is formalized as the treewidth or
pathwidth It is easy to nd examples of families of graphs with b ounded treewidth
such as the sets of series parallel graphs Halin graphs outer planar graphs and
graphs with bandwidth at most k see Section and Bo d a One also nds
other natural classes of graphs with b ounded treewidth in the study of the reliability
of communication networks and in the evaluation of queries on relational databases
MM Bie Arn The treewidth parameter also arises in places such as the ef
ciency of doing Choleski factorization and Gaussian elimination BGHK DR
The pathwidth and treewidth of graphs haveproven to b e of fundamental imp or
tance for at least two distinct reasons their role in the deep results of Rob ertson
and Seymour RS RS a RS a RSc and they are common denominators
leading to e cient algorithms for b ounded parameter values for many natural input
restrictions of NP complete problems To further explain this second phenomenon
consider the classic NP complete problem of determining if a graph G hasavertex
cover of size k where b oth G and k are part of the input It is known that if k is
xed i e not part of the input then all yes instances to the problem have path
width at most k This restricted problem can b e solved in linear time for any xed
k see Chapter For more examples b esides the b ounded width families studied
in this dissertation see Bo d a FL M oh In general many problems such
CHAPTER BOUNDED COMBINATORIAL WIDTH
as determining the indep endence of a graph can b e solved in linear time when the
input also includes a b ounded width path decomp osition or tree decomp osition of
the graph see Arn AP Bo d a CM WHL and Bo d a for many further
references Examples of this latter typ e are di erent from the vertex cover example
b ecause a width b ound for the yes instances is not required e g for every xed k
there exists a graph of indep endence k with arbitrarily large treewidth
Another well known fact ab out the set of graphs of pathwidth or treewidth at
most k is that each set is characterizable by a set of forbidden minors We denote
these parameterized lower ideals by k Pathwidth and k Treewidth For ex
ample some recently discovered obstruction sets for Pathwidth Treewidth
Treewidth are shown in Figures and on page at the end of this
chapter APC Besides the single obstruction K for Treewidth a set of
obstructions for Pathwidth see Kin is the only other known characterization
for b ounded pathwidth or treewidth
Treewidth and k trees
Informally a graph G has treewidth at most k if its vertices can b e placed with
rep etitions allowed into sets of order at most k and these sets arranged in a tree
structure where for eachedge u v of G b oth vertices u and v are members of
some common set and the sets containing any sp eci c vertex induces a subtree
structure Figure illustrates several graphs with b ounded treewidth pathwidth
Below is a formal de nition of treewidth
De nition A treedecomposition of a graph G V E is a tree T together
with a collection of subsets T of V indexed bythevertices x of T that satis es
x
T V
x
x T
For every edge u v of G there is some x suchthat u T and v T
x x
If y isavertex on the unique path in T from x to z then T T T
x z y
CHAPTER BOUNDED COMBINATORIAL WIDTH
Figure Demonstrating graphs with b ounded treewidth and pathwidth
CHAPTER BOUNDED COMBINATORIAL WIDTH
The width of a tree decomp osition is the maximum value of jT j over all vertices
x
x of the tree T A graph G has treewidth at most k if there is a tree decomp osition of
G of width at most k Path decompositions and pathwidth are de ned by restricting
the tree T to b e simply a path see Section b elow
De nition A graph G is a k tree if G is a k clique complete graph K or G
k
is obtained recursively from a k tree G by attaching a new vertex w to an induced
k clique C of G such that the op en neighborhood N w V C A partial k tree is
any subgraph of a k tree
Alternatively a graph G is a k tree if
G is connected
G has a k clique but no k clique and
every minimal separating set is a k clique
The ab ove result was taken from one of the manycharacterizations given in Ros
The next useful result app ears in many places ACP RS a Nar
Theorem The set of partial k trees is equivalent to the set of graphs with
treewidth at most k
Pathwidth and k paths
For our representations of graphs of b ounded width we use graphs that havea dis
tinguished set of lab eled vertices These are formally de ned next
De nition For a p ositiveinteger k a k simplex S of a graph G V E isan
injectivemap f kg V Ak boundariedgraph B G S is a graph G
S
together with a k simplex S for G Vertices in the image of are called boundary
S
vertices often denoted by The graph G is called the underlying graph of B
CHAPTER BOUNDED COMBINATORIAL WIDTH
Wenow de ne a family of graphs where each memb er has a linear structure As
in Theorem a graph has pathwidth at most k if and only if it is a subgraph
of an underlying graph in the following set of k b oundaried graphs Our de nition
of a k path is consistent except for the base clique with other de nitions that use
a set of b oundary vertices with one fewer vertex see for example BP Later in
Lemmas and we justify these two statements
De nition A graph G is a k path if there exists a k b oundaried graph
B G S in the following family F of recursively generated k b oundaried graphs
K S F where S is any k simplex of the complete graph K
k k
If B V E S F then B V E S F
where V V fv g for v V and for some j f kg
a E E f i v j i k and i j g and
S
v if i j
i b S is de ned by
S
i otherwise
S
A partial k path is subgraph of a k path
Item in the ab ove de nition states that a new b oundary vertex v can b e added
as long as it is attached to any current k simplex within the active k simplex S
The set of b oundary vertices of B is the closed neighborhood N v of the new vertex
v
The de nition of a tree decomp osition is easily sp ecialized for this path structured
case
De nition A path decomposition of a graph G V E is a sequence X X X
r
of subsets of V that satisfy the following conditions
X V
i
i r
For every edge u v E there exists an X i r suchthatu X and
i i
v X i
CHAPTER BOUNDED COMBINATORIAL WIDTH
For i j k r X X X
i k j
The pathwidth of a path decomposition X X X is max jX j The path
r i r i
width of a graph G denoted pw G is the minimum pathwidth over all path decom
p ositions of G
The next de nition makes the development of b ounded pathwidth algorithms
easier to understand and prove correct
De nition A path decomp osition X X X of pathwidth k is smooth if
r
for all i r jX j k and for all i r jX X j k
i i i
It is evident that any path decomp osition of minimum width can b e transformed
into a smo oth path decomp osition of the same pathwidth There is a corresp onding
notion of a smooth treedecomposition See Bo d a for a simple pro cedure that
converts any tree decomp osition into a smo oth tree decomp osition From a smo oth
decomp osition of width k we can easily emb ed a graph in a k path or k tree e g see
pro of of Lemma
To initiate the reader and for completeness wenowgivetwo basic results
regarding the notion of pathwidth RS a We use part of the second result when
weprove that our algebraic graph representation given in Section is correct
Lemma If a graph G has comp onents C C C then
r
pw G max fpw C g
i
i r
i i i
Pro of For i r letX b e a path decomp osition for comp onent C X X
i
s
i
with minimum width w pw C The sequence of sets
i i
r r r
X X X X X X X X X
s s s
r
S S
r r
is a path decomp osition of G since V C V G E C E G
i i
i i
a b
and X X for i s j s and a b r The width of
a b
i j
r r
this decomp osition is max fw g so pw G max fpw C g
i i
i i
CHAPTER BOUNDED COMBINATORIAL WIDTH
Let X X X b e a path decomp osition of G of minimum width For any
s
comp onent C of G letD X V C forj s We claim that D D D
i j j i s
is a path decomp osition of C Since for all v V G there is an X suchthatv X
i k k
S
D V C Likewise since for every edge u v E G there is an X such
j i k
j s
that u X and v X So if u v E C then b oth u D and v D
k k i k k
Finally for i j k s X X X implies D D D Therefore since
i k j i k j
jD j jX j for k s pw C pw G forany comp onent C of G
k k i i
The following well known lemma shows that the pathwidth of a partial k path is
at most k Thus we can justi ably think of partial k paths as either subgraphs or as
minors of k paths
Lemma If G is a k path then pw G k Further if H is a minor of G H G
m
then pw H k
Pro of We rstprove that any k path G obtained from a de ning k b oundaried
graph G S has a path decomp osition X X X of width k where X imag e S
r r
i e X is the set of b oundary vertices For the base case of G a k clique the
r
hyp othesis holds since X f k g V G is a path decomp osition of width
k For the inductive step assume X X X is a path decomp osition for G Let
r
path built from some G S by applying case of De nition If weset G b e the k
X imag e S thenX X X X is the required path decomp osition of
r r r
G The new vertex v is in X all new edges u v in E G n E G have
r
u X and v X and X X X X implies X X X for
r r i r i r i r j
i j r
To see that any k path G has a lower b ound of k for its pathwidth rst notice that
G contains a clique C of size k For any path decomp osition X X X X
r
of G let Y X V C for i r Ifany jY j k then the width of X is at least
i i i
k Supp ose jY j k for all Y There must then b e a Y and a Y i j r such
i i i j
that u Y n Y and v Y n Y for di erentvertices u and v Since u v E C
i j j i
there must b e some Y k r suchthat u Y and v Y Now fu v g Y Y
k k k i j
so either k ior k j Butv Y for k i and u Y for k j This contradicts
k k
X b eing a path decomp osition i e it fails the edge covering requirement Therefore
the pathwidth of any k path is k
CHAPTER BOUNDED COMBINATORIAL WIDTH
For the second part of the theorem we start with a path decomp osition X
X X X of width k for a k path G It is su cienttoshow that any one step
r
minor H of G has a path decomp osition of width at most k For edge deletions
H G nf u v g the original path decomp osition X is a path decomp osition of
the minor H For isolated vertex deletions H G nfv g the path decomp osition
X nfv g X nfv g X nfv g is a path decomp osition of the minor H For edge
r
contractions H G u v let
X if X fu v g
i i
Y
i
X fw gnfu v g otherwise
i
for i r where w is the new vertex created by the contraction The sequence
Y Y Y Y is a path decomp osition of the edge contracted minor H The
r
widths of these path decomp ositions for the three typ es of one step minors is at most
k
We justify the edge contraction case and leave the other two simpler cases to the
S
Y V H Let a b E H If fa bg fu v g then reader By de nition
i
i s
some Y X contains b oth a and b Otherwise consider any edge a b x w An
i i
edge incidenttovertex w implies that either x u E G or x v E G Thus
some X contains x and either one or b oth of u and v And so Y X fw gnfu v g
i i i
contains b oth x and w Finally let i b e the minimum index suchthat w Y and k
i
b e the maximum index such that w Y Without loss of generality assume u X
k i
If u X then Y X nfv g and Y is a path decomp osition of H Otherwise
k
X and v X Since u v E G there exists a j i j k suchthatboth u
j k
v X Since X is a path decomp osition u X for i m j and v X for
j m m
j m k Sow Y for i m k and w Y for m i or m k Therefore
m m
for i j k s wehave Y Y Y
i k j
k
Corollary Any graph of order n and pathwidth k has at most n k
k edges
y induction on the number Pro of We easily prove that equality holds for k paths b
of vertices Since the smallest k path is the complete graph G K our base case
k
k
holds since jE K j If a k path G has n k vertices then there
k
CHAPTER BOUNDED COMBINATORIAL WIDTH
exists a vertex v that was added to a smaller k path G G nfv g using the recursive
de nition of k paths When vertex v was added to G there were exactly k new edges
k k
added to o So G must have n k k k n k k
edges
From the ab ove corollary we see that graphs of b ounded pathwidth haveatmost
a linear numb er of edges with resp ect to n It is not di cult to show that graphs
of treewidth k also have this same bound So since the input size is linear in the
number of vertices for our b ounded width algorithms there is no confusion when
we talk ab out having a linear time algorithm
Wenow complete this subsection by proving the converse of Lemma
Lemma Any graph G of pathwidth at most k is a subgraph of a k path
Pro of If G hasatmostk vertices wecanembed G in the base clique K
k
the smallest k path Otherwise let X X X b e a smo oth path decomp osition
r
of G of width k We can map the vertices of X into the base clique K By
k
i
induction assume that wehaveemb edded the vertices of X in a k path where
j
j
X is the set of b oundary vertices of the k b oundaried graphs B G S
i i i i
generated by the rules of De nition We can construct the next k b oundaried
graph B G S by adding the unique vertex v X n X in step of
i i i i i
De nition and dropping the unique vertex v X n X from the simplex S of the
i i i
previous k path G Since each X i r is mapp ed to a k clique every edge
i i
of G is mapp ed to an edge of the k path G G
r
Other graph theoretical widths
There are actually manytyp es of combinatorial width metrics for graphs Often
a set of graphs b ounded by one typ e of combinatorial width is a subset of a class
of graphs with another b ounded parameter A relationship b etween treewidth and
several graph widths is given by Bo dlaender in Bo d b In addition Wimer s disser
tation Wim b explores the relationships b etween terminal recursive families e g
series parallel graphs outer planar graphs and cacti This work is now highlighted in Figure
CHAPTER BOUNDED COMBINATORIAL WIDTH
treewidth k
k outerplanar chordal graph max clique k
cutwidth k
series parallel
pathwidth k
outerplanar
cyclic bandwidth k
Halin graphs
bandwidth k
interval graph max clique k
almost k trees
prop er interval graph max clique k
cacti
forests
caterpillars
trees
Note that each constants k denotes a di erent constant For example
the set of all graphs of bandwidth k is contained in the set of graphs of
cutwidth k k k
Figure A partial order of several b ounded width families of graphs
CHAPTER BOUNDED COMBINATORIAL WIDTH
We brie y describ e two of the b ounded width families given in Figure that
originated from VLSI linear layout problems A linear layout of a graph G V E
is a bijection f V f jV jg that is an assignmentofintegers to jV j
to the vertices The bandwidth of a layout is the maximum value of jf i f j j
over all edges i j E that is the maximum distance any edge has to span in
the layout The cutwidth of a layout is the maximum number of edges i j such
that f i k f j over all k jV j that is the maximum numb er of lines
that can b e cut b etween anyla yout p osition k and k The bandwidth cutwidth
of the graph is the minimum bandwidth cutwidth over all linear layouts Both of
these min of max de nitions are similar to the pathwidth and treewidth de nitions
that wesaw earlier In fact the pathwidth of a graph is equivalent to the vertex
separation of a graph which is also de ned as a linear layout problem see Chapter
The de nition of interval graphs may b e found in most intro ductory graph theory
b o oks they are also men tioned in Chapter We conclude this section with a
description for two of the other less known b ounded width families A Halin graph
is a planar graph G T C where T is a tree of order at least with no degree
twovertices and C is a simple cycle connecting all and only the leaves of T Acactus
member of the cacti family is a connected outer planar graph that has single edges
and simple cycles as its blo cks
Algebraic Graph Representations
Recall that we are mainly interested in nite simple graphs However some of our
graphs haveavertex b oundary of size k meaning that they have a distinguished set
of vertices lab eled k We sometimes use instead of k as a b oundary lab el
One imp ortant op eration on k b oundaried graphs is giv en next
De nition Two k b oundaried graphs each with a b oundary of size k can b e
glued together with the op erator called circle plus that simply identi es vertices with the same b oundary lab el
CHAPTER BOUNDED COMBINATORIAL WIDTH
Example The binary op erator on two b oundaried graphs A and B is il
lustrated b elow Note that common b oundary edges in this case the edges b etween
b oundary vertices and are replaced with a single edge in G A B
G A B
A B
Op erator sets and t parses
In this section weshowthat t b oundaried graphs of pathwidth at most t are
generated exactly by strings of unary op erators from the following operator set
t
V E
t t
n n
V f t g and
t
ij j i j tg E f
t
To generate the graphs of treewidth at most t the additional binary op erator is
addedto The semantics of these op erators on t b oundaried graphs G and
t
H are as follows
n
G i Add an isolated vertex to the graph G and lab el it as
the new b oundary vertex i
ij Addanedgebetween b oundary vertices i and j of G and G
ignore if the edge already exists
G H Take the disjoint union of G and H except that b ound
ary vertices of G and H that are lab eled the same are
identi ed i e the circle plus op erator
De nition A parse is a sequence or tree if is used of op erators g g g
n
n n
in that has vertex op erators i and j o ccurring in g g g b efore the
n
t
ij i j t rst edge op erator
CHAPTER BOUNDED COMBINATORIAL WIDTH
De nition A t parse is a parse g g g in where all the vertex op er
n
t
n n n
ators t app ear at least once in g g g That is a t parse is a
n
parse with t b oundary vertices
For clarity wesaythatatreewidth t parse is any t parse containing at least one
op erator and a pathwidth t parse is any t parse without op erators
Intuitively t parses represent t b oundaried graphs that havepathwidth or
treewidth at most t We justify this statement later in this section These graphs are
constructed by using the rules de ned for the unary vertex and edge op erators and
the binary circle plus op erator The simplex S of a t parse i e the corresp onding
n
b oundaried graph is informally de ned to b e i i i t where each
S
n
i is the last o ccurrence in the parse The symbol is used to denote the set of
b oundary vertices of a b oundaried graph
As we pro ceed with our obstruction set computation theory we use the phrase
a t parse G in one of three ways
as a t b oundaried graph also called G
as an implied path or tree decomp osition of a graph or
as a data structure e g for dynamic programs
Thus a t parse G is used with standard ob ject oriented terminology i e a t parse
G is a b oundaried graph and G has a decomp osition where any op erations on
graphs may also b e applied to a t parse G
De nition Let G g g g bea t parse and Z z z z beany
n m
sequence of op erators over Theconcatenation ofG and Z is de ned as
t
G Z g g g z z z
n m
is called an extension For The t parse G Z is called an extended t parse and Z
t
the treewidth case G and Z are viewed as two connected subtree factors of a parse
tree G Z instead of two parts of a sequence of op erators
CHAPTER BOUNDED COMBINATORIAL WIDTH
n
Avertex op erator g i forany i t ina t parse G g g g is
j n n
n
called a boundary vertex if g i for j k n
k
De nition A graph G is generated by if there is a t b oundaried graph
t
B S G g g g suchthat B G for some t parse G That is G is
n n n
isomorphic to an underlying graph of a t parse
For ease of discussion throughout the remaining part of this chapter we limit
ourselves to graphs of b ounded pathwidth while leading up to our obstruction set
searchtheory In certain situations where required weprovide additional information
p ertaining to graphs of b ounded treewidth
De nition Let G g g g b e a b oundaried graph represented as some
n n
parse over Apre x graph of length m of G is G g g g m n
t n m m
Lemma Every t parse G g g g represents a graph with a path de
n n
comp osition of width t
Pro of Without loss of generality letG g g g bea t b oundaried
n n
graph Let v denote the index of the i th vertex op erator in the t parse G LetI
i n
b e the set of these v and for i I
i
fj j g isaboundaryvertex of G g X
v v i
j i
We claim that X X X X is a path decomp osition of width t for G
n
jI j
For anyvertex op erator g de ne l abel g tobei if g is the i th vertex op erator
k k k
S
in G By de nition of the X s X f jI jg V G For anyedge
n i i n
ij of G letb g andb g denote the indices of the preceding op erator g
n k k k
n n
vertex op erators i and j resp ectively For each edge op erator g ij of G
k n
let u l abel b g and v label b g Since edge op erators add edges to the
k k
current b oundary of a pre x graph either X or X must contain b oth u and v
u v
Assuming i j X X represents the b oundary vertices of G that did not change
i j v
j
from G Soforany k i k j wehave X X X X This implies that
v i j i k
i
X X X
i j k
CHAPTER BOUNDED COMBINATORIAL WIDTH
n n n
Finally since only and exactly the vertex op erators t app ear
in G wehavemax fjX jg t So X is a path decomp osition of width t for
n i
i jI j
G
n
And next weprove the result in the other direction
Lemma Every partial t path of order at least t is represented by some
t parse
Pro of We rst show that every t path can b e represented by some t parse We prove
inductively from the recursive de nition of t paths The initial t clique can b e
represented by the following sequence of op erators over
t
n n n n n
t t t t t
For the inductive step assume that any t path G with an active t simplex
S is represented by an op erator string G g g g suchthat is de ned
m m S
prop erly The recursive op eration of replacing a simplicial vertex with a new vertex
v forming a new clique can b e mo deled by app ending the op erator sequence of the
form
n
i i i i ii it
to G dep ending on which simplicial vertex b oundary vertex i should b e replaced
m
Thus any t path can b e represented byat parse over
t
To representpartial t paths note that each edge is represented by one or more
edge op erators By simply deleting all of these edge op erators any edge in the t parse
is removed Thus by rep eatedly deleting edges wegeta t parse for any partial t path
Note that many t parse representations may exist for a partial t path since many
path decomp ositions are generally p ossible Finally combining the previous two
lemmas wehave the following theorem
Theorem The family of graphs t parses generated by equals the family of
t
partial t paths of order at least t
CHAPTER BOUNDED COMBINATORIAL WIDTH
Corollary A graph G has pathwidth t if and only if it is generated by but
t
not
t
Pro of If a graph G is generated by then it has pathwidth t If G has
t
pathwidth t then it is a partial t path and by the ab ove theorem it has a t parse
representation over
t
For a treewidth analog to Theorem wehave the following result
Theorem The set of treewidth t parse s represents the set of graphs of order at
least t and treewidth at most t
Pro of We rst show that any t parse G has a tree decomp osition T fT g ofwidth
x
t where the current b oundary of G is in some T x T We provethisby induction
x
on the number of operators in G and whether or not G has any op erators For
the base cases of no op erators wehaveby Lemma a path decomp osition
X X X of width t Thus G has a tree decomp osition P fX j i V P g
r r i r
where P denotes a path of length r Note that by our constructive pro of the set X
r r
contains the current b oundary of G
Now assume G contains at least one op erator Wehave three cases If G
G G then let T fT g and T fT g b e tree decomp ositions for G and G
x x
resp ectively Since b oth G and G have fewer op erators than G there exists sets T
x
that contain the b oundary of G and G or of G itself A tree decomp osition and T
x
of G of the desired form is constructed byidentifying T and T in the two subtree
x x
g b e a tree decomp osition of ij then let T fT decomp ositions If G G
x
the desired form for G Since b oth b oundary vertices i and j are in some vertex set
T for some x T T fT g is the required tree decomp osition for G Similarly if
x x
n
G G i let T fT g b e a tree decomp osition of the desired form for G Let
x
T b e the tree constructed byattaching a vertex i to T at vertex x whereT contains
x
ts the old all the b oundary vertices of G SetT T nfi g fig where i represen
i
x
b oundary vertex This proves that every t parse has treewidth at most t
Nowweshowthatany t tree G with at least t vertices contains a t parse
representation As in the pro of of Lemma any partial t tree is representable
CHAPTER BOUNDED COMBINATORIAL WIDTH
by removing the appropriate edge op erators Our pro of is based on a smo oth tree
decomp osition T fT g of G In particular jT j t for all x T and jT T j t
x x u v
whenever u and v are adjacentinT We recursively build a t parse for G by arbitrarily
picking a ro ot vertex r of T and having T as the nal t parse b oundary
r
We rst require a function that takes a t parse H and returns a t parse
i j
H where the b oundary lab els i and j have b een interchanged The reader should
observe that the function is easy to construct For the pro of b elowwe assume
i j
that the underlying graph G is lab eled jGj and that any partial t parse will
have b oundary vertex i less than b oundary j whenever the underlying lab el for i is
less than the underlying lab el for j Thisboundary order is needed to keep things
aligned when we use the op erator
If G has t vertices i e a clique K then the construction given in Lemma
t
for the t path case is su cient Wenow recursively build a t parse for G based on
the degree d of a ro ot vertex r Letv v b e the neighb ors of vertex r in
d
i i
T For each i d consider the subtree decomp osition T fT g induced bythe
x
G G for these pieces subtree of T ro oted at v By induction wehave t parses
d i
of G with b oundary sets T T resp ectively
v v
d
If d wehave the following t parse where i denotes the b oundary vertex
representing the single vertex in T n T for the original t parse G using the pro of
v r
metho d of Lemma
n
G i i i i ii it
If the new b oundary vertex i is out of b oundary order with resp ect to T then wecan
r
apply the function p ossibly several times with di erentvertices i and j
i j
If d then we apply one vertex op erator for each subtree parse and a total of
d circle plus op erators to combine the pieces The vertex op erator is chosen the
same wayasinthed case and wemayhave to p ermute with the function
i j
Let G denote a particular t parse for i d The t parse for G is then created
i
by the following parse
CHAPTER BOUNDED COMBINATORIAL WIDTH
G
d
G
d
G
G G
Note that multi edges created by the rep eated op erators are ignored
Some t parse examples
The next three examples illustrate our algebraic representation of t b oundaried graphs
of b ounded width The rst two t parse examples are graphs of pathwidth which
arethencombined with the circle plus op erator to form a graph of treewidth
Example A t parse with t and the graph it represents The shaded
vertices denote the nal b oundary
n n n n n n
Example Another parse and the graph it represents
CHAPTER BOUNDED COMBINATORIAL WIDTH
n n n n n n n
Example We demonstrate the circle plus op erator with the b oundaried
graphs parses given in the previous two examples The second graph is re ected
and glued onto the rst graph s b oundary In general the pathwidth of a t parse
usually increases when the binary op erator is used although not in this example
Other op erator sets
Our t parse representation for graphs of pathwidth or treewidth at most t is not
unique Wenow presenttwo di erent op erator sets for representing graphs of b ounded
combinatorial width Many other algebraic representations are available sometimes
in disguised form by other sources e g see Bor BC CM Wim b The
following two examples illustrate two extremes regarding the numb er of op erators
required to represent graphs of pathwidth or treewidth of at most k The rst set
shows that only a constantnumb er of op erators is needed indep endentofthe
width k Fel The second set uses a p olynomial numb er of op erators p er b oundary
CHAPTER BOUNDED COMBINATORIAL WIDTH
size but generates k b oundaried graphs for pathwidth k instead of graphs with
b oundary size k Wepoint out that another middle ground op erator set based
on a combination of these two sets is given in Lu
A constant sized op erator set for graphs with pathwidth at most k consists
k
of two b oundary p ermutation op erators p and p and the two t parse graph building
n
op erators and The domain for these op erators is again k b oundaried
graphs The p ermutation op erators are used to relab el the b oundary These p er
mutations given in the standard cyclic form are p and p k
These two p ermutations generate the symmetricpermutation group S and hence
k
by applying in succession arbitrary relab elings of the b oundary are p ossible It is
easy to see that with its op erators generates exactly the same set of b oundaried
k
graphs as our pathwidth t parse op erator set We also observe the following
k
Observation A graph G has treewidth at most k if and only if it is obtainable
by using the op erators f g
k
A big pathwidth op erator set called fork b oundaried graphs of pathwidth at
k
most k contains the t parse op erator set and has the following two additional
k
op erator typ es In the de nitions b elow G denotes any k b oundaried graph
i Add a p endentvertex to the current b oundary vertex i G
of G and lab el it as the new b oundary vertex i
G fb b b g Add a new interior vertex and attach it to all of the
p
b oundary vertices b b b of G p k
p
Example Belowwe illustrate the op erator set in generating a b oundaried
graph of pathwidth
n n n n
f g f g
CHAPTER BOUNDED COMBINATORIAL WIDTH
Wenow prove that this op erator set that uses a smaller b oundary size can b e
used to represent graphs of b ounded pathwidth
Theorem The op erator set generates precisely the set of graphs of path
k
width at most k
Pro of Clearly if a graph has n k vertices which means it has pathwidth at
most k then it is representable by a sequence that starts with n distinct vertex
op erators and an edge op erator for each edge
We next show without loss of generality thatevery well de ned sequence
n n
S k s s s
r
of op erators represents a graph of pathwidth at most k We do this by showing
k
how to build a t parse G t k for representing the underlying graph For each pre x
S s s s wehave a corresp onding pre x t parse G of G where
i i
i
is some increasing integer function The construction uses a spare but variably
lab eled b oundary vertex of for simulating the fb b b g and i op erators
k p
For the following discussion weuse j to denote a injective map from the b oundary
of S to the b oundary of G and the integer v to denote the unique b oundary lab el
i
i
n n
not in the image of We start by setting G k to b e the
identity map and v k Nowwehave four cases for the inductivesteps
n n
If s i then for i i G G i
r
r r
ij then G G i j If s
r r r
n
If s i then G G v i v Swap the values of i and v
r
r r
If s fb b b g then
r p
n
G G v b v b v b v
r r p
Thus since any pathwidth t parse t k has pathwidth at most k so do es any
k b oundaried graph generated by k
CHAPTER BOUNDED COMBINATORIAL WIDTH
Wenow show that any k path G is representable by a string of op era
k
tors Again this is su cient since we can remove edge op erators ij or replace
fb b b g op erators to obtain any partial k path Let X X X X be a
p r
smo oth path decomp osition of G We constructively build a partial parse P using
i k
S
op erators for eachvertex induced k path de ned by X The current b oundary
j
j i
of P will b e P X X for i r For eachvertex u G that is on
i i i i
the current b oundary of P de ne u to b e the corresp onding b oundary lab el in
i i
k The initial K clique is parsed by
k
n n n n
P
k k k k k f k g
where u is assigned arbitrarily for the vertices X n X Letol d denote the unique
i
vertex in X n X and new denote the unique vertex in X n X for i r
i i i i i
Wehavetwo cases to consider when constructing P from P for i r
i i
If new ol d then
i i
P P f k g
i i
else i e new ol d for j ol d
i i i i
P P j j j j jj jk
i i
then new j The nal parse for G is simply P P f k g
i i r r
to end with a K clique assuming r
k
One of the reasons whywepicked our t parse op erator set over other p ossible
t
sets is that wewant each unary op erator to add something signi cant but not to o
complex to its t b oundaried graph argument That is wewant a smo oth and
rapid path via these graph building op erators to the obstructions but do not want
to oversho ot the graph family to o far by adding complex graph pieces That is we
believe that the search tree is smaller using our choice of op erators An analogy from
the computer architecture world is that wewould prefer a RISC chip over one with a
full and p owerful instruction set or in the other extreme one with only tedious nano
co de primitives Another reason for our choice is that wewant a pre x prop ertyofthe
parse strings This prop erty for obstruction set searching is elab orated in Chapter
CHAPTER BOUNDED COMBINATORIAL WIDTH
Simple Enumeration Schemes
Tohelptackle the apparently intractable task of nding obstruction sets weneed
to limit the domain of the search If we know that a particular obstruction set O is
nite any graph invariantover memb ers of O will have an upp er b ound To take
advantage of a particular b ounded invariant a metho d is needed to generate all these
restricted graphs This set of graphs must b e substantially smaller than the set of
graphs of order at most maxfjO j O Og As indicated by the topic of this
i i
chapter the two main invariants that we exploit are the pathwidth and the treewidth
metrics
As seen by the examples in Section it is easy to represent with a computer
graphs of pathwidth or treewidth of at most t by strings of unary op erators or by trees
with the additional binary op erator This suggests a natural wayofenumerating
all these b ounded width graphs Just enumerate all p ossible valid combinations of
t parse strings or t parse trees Unfortunately many di erent t parse s corresp ond
to the same underlying graph To reduce our search pro cess we need to knowwhen
t wo t parses represent the same graph We formalize this concept next
De nition Two k b oundaried graphs B G S andB G S are
free boundary isomorphic denoted B B if there exists a graph isomorphism
f
between G and G such that
f i j i k g f i j i k g
S S
That is b oundary vertices of G are mapp ed under to b oundary vertices of G
And B and B are xed boundary isomorphic if there exists an isomorphism such
that
i i for i k
S S
That is each b oundary vertex i of G is mapp ed under to b oundary vertex i of G
Our goal is to nd an enumeration scheme that generates each free b oundary
isomorphic t parse at most once
CHAPTER BOUNDED COMBINATORIAL WIDTH
Canonic pathwidth t parse s
One simple way of generating each free b oundary isomorphic t parse is to de ne
equivalence classes of t parses and generate one representative for each class of free
b oundary isomorphic graphs A t parse is said to b e canonic with resp ect to some
linear ordering of t parse s if it is a minimum t parse within its equivalence class For
an enumeration scheme these minimum representatives should b e de ned so that
extending the set of canonic representatives of order n generates a set or sup erset of
the canonic representatives of order n Here ifwe generate a non canonic t parse
of order n we discard it b efore generating the canonic representatives of order
n We call an enumeration order scheme simple if this prop erty holds
The simplest linear ordering of t parses is a lexicographical order of the parse
strings We de ne the lex canonic order between two t parse s by comparing op erators
at equal indices from the b eginning of each string until a di erence is found The
individual op erators in are related as follows
t
n
Vertex op erator i is less than any edge op erator jk
n n
Vertex op erator i is less than anyvertex op erator j whenever i j
Edge op erator ij where i j is less than edge op erator kl where k l
whenever i k or i k and j l
A canonic t parse in the lex canonic order called a lex canonic t parse is
l
any t parse G such that G H for any free b oundary isomorphic t parse H G
l
f
where H G Since any pre x of a lex canonic t parse is also lex canonic a simple
enumeration scheme is p ossible using this form of canonicity This pre x prop ertyis
easily seen by assuming that a pre x P of a lex canonic t parse G P S is not lex
canonic then a contradiction arises regarding G b eing lex canonic That is consider
a lexicographically less t parse P S G that is free b oundary isomorphic to G
l
where P P and P P
l
f
Example Several t parses are listed b elow in increasing lex canonic order The
fourth t parse is not lex canonic since it is free b oundary isomorphic to the second
CHAPTER BOUNDED COMBINATORIAL WIDTH
t parse Likewise the fth t parse is free b oundary isomorphic to the rst whichis
also lex canonic
n n n n
n n n n
n n n n n
n n n n
n n n n
Wenow present an alternate canonic representation for pathwidth t parses The
main b ene t of this scheme based on a non lexicographical order is that most non
canonic t parses are easily determined bychecking for required canonicity prop erties
of the parse string For any t parse P letvseq P bethevertex subsequence of
n
the parse that is just the vertex op erators i for some i and let vpos P bethe
p ositions of the vertex op erators in the original t parse
Example For the t parse
n n n n n n
G
wehave
n n n n n n
v seq G
and
vpos G
De nition For two t parses P and P of the same free b oundary graph we de
ne a linear order as follows where the symbol denotes the lexicographical order
c
on integer sequences and denotes the lex canonic order on op erator sequences
l t
If vpos P v pos P thenP P
c
vseq P thenP P Else if v seq P
l c
Here v pos P v pos P
Else if P P then P P
l c
Here v pos P v pos P and vseq P vseq P
CHAPTER BOUNDED COMBINATORIAL WIDTH
The order is a linear order b ecause is a linear order i e if case is ever
c l
reached then either P P or P P holds
c c
For the remainder of this section a t parse P of a free b oundary graph G is
termed canonic if there is no other parse P such that P P and P P The
c
f
idea b ehind this t parse order is that wewantvertex op erators to come earlier in the
n
parse For the linear order wenotethatanyvertex op erator i and anyedge
c
op erator jk do not need to b e compared lexicographically
Wenow state some useful facts ab out canonic t parses using the order
c
n
Lemma Let G g g g b e a canonic t parse If g i and g
n n k k
n
ab k k j are consecutivevertex op erators then for any edge op erator g
k
k either a j or b j
Pro of Assume that for some k k k g ab where a j and b j Clearly
k
the following parse G represents the same free b oundary graph
n
n n
ab g g G g g g i g g g j g
k n k k k k k
n
But vpos G is lexicographically less than v pos G This can not happ en since G
n n
n
is canonic So wemust haveeithera j or b j in order to prevent the p ossible
edge shift
The previous lemma states that any edge op erators that immediately precedes a
n n
vertex op erator i must b e adjacent to the previous i The next result regarding
the p osition of the b oundary edges follows easily from the previous lemma
Lemma Let G be a canonic t parse If g ij is a b oundary edge of G
n m n
m n then there are no vertex op erators in p ositions m m n
n n
Pro of If not the next vertex op erator would havetobe i or j
The following lemma states that any pre x of a canonic t parse is also canonic
This means that using to de ne t parse canonicity is suitable for a simple t parse
c
enumeration scheme
CHAPTER BOUNDED COMBINATORIAL WIDTH
Lemma If G is a canonic t parse then the pre x G is a canonic t parse
n n
Pro of If this is not true then there exists H G withH G Let
n n n c n
f
V G V H b e a free b oundary isomorphism mapping b etween G
n n n
and H De ne
n
n n
if g i
n i
h
n
if g ij
i j n
Now consider H H h The parse H is isomorphic to G with H G So
n n n n n n c n
wemust have H G
n c n
Wenow turn to the problem of determining when an extension of a canonic
t parse is also canonic
Lemma Let G b e a canonic t parse If Z ij is a single edge op erator
n
extension then G G Z can b e tested for canonicity with a constantnumber
n n
of isomorphism calls
Pro of Supp ose G is not canonic Then there exists a free b oundary isomorphic
n
t parse H G with H G Since b oth G and H contain
n n n c n n n
f
b oundary edges weknow from Lemma that the last vertex op erator has index
k n that is k n numb er of b oundary edges An isomorphism mapping
V G V H shows that the t parses H and G are free b oundary
n n k k
G nf a b g isomorphic This is seen by noting that for anyedge a b of G
n n
f
H nf g Since G is canonic the pre x G is also canonic likewise H so
n a b n k k
H G implies H G
k k k k
f
Thus if G is not canonic then its canonic t parse representation is identical to
n
G except for the b oundary edges at the end of the parse If there are i b oundary
n
t
edges then wehave at most p ossible t parse s to consider
i
Example Below is an instance of where a constantnumb er of isomorphism calls
would tell us that the extended t parse H is not canonic This example is created
by adding an edge b etween a p endentvertex and an isolated b oundary vertex of the
CHAPTER BOUNDED COMBINATORIAL WIDTH
canonic t parse G The t parse K is free b oundary isomorphic to H and canonic
n n n n n
G
n n n n n
H
n n n n n
K
We can easily eliminate an isomorphism check for several of the p ossible t parses
mentioned in the previous lemma For a prosp ective t parse H to b e free b oundary
n
isomorphic to G the degree sequences of the b oundaries must b e identical These
n
degree sequences can b e re ned to include b oth the numb er of b oundary and non
b oundary incident edges That is these two ordered pair degree sequences one for
G and one for H must coincide
n n
n
Observation Let G be a canonic t parse If Z i a single vertex op erator
n
extension then G Z is non canonic if Lemma is violated whichislikely and easy
n
to check
The next lemma helps us detect other non canonic situations for any t parse of
length n that ends with a vertex op erator
Lemma With m nb eing the smallest index of any edge op erator of a canonic
t parse G there are no consecutivevertex op erators g and g in G for m i n
n i i n
n
o identical vertex op erators i consecutivein G One of Pro of First consider tw
n
n
these can b e replaced bya and moved to the rst of the string since the semantics
n n
of i i causes an internal isolated vertex Nowa su xofG with two di erent
n
consecutivevertex op erators can b e rewritten as assuming G is canonic
n
n n n n n
G j ik ij i j ij j ik i
n c
that is less in the order If there are more than two consecutivevertex op erators
c
n n n
ab i j k thenwe can also shift one of these earlier Here the vertex
that can b e shifted b efore the edge op erator ab is determined by fi j k gnfa bg
CHAPTER BOUNDED COMBINATORIAL WIDTH
One thing that is not quite resolved is how to e ciently handle the not so easy
vertex op erator cases If b oth Lemmas and are not violated we still do not
n
knowifat parse G i is canonic Webelieve that a non canonic t parse can pass
n
b oth these lemmas for only a very few sp ecial cases mayb e not even enough times
to worry ab out Even without a fast canonic algorithm for this case we still havea
fast metho d for generating all b ounded pathwidth graphs Since the set of graphs of
pathwidth at most t is obtained from the set of canonic t parse s and we generate a
sup erset the ab ove statements provide us with an e cient means of generating each
such partial t path with a constant sized b oundary exactly once If one wishes a
free b oundary isomorphism algorithm can b e used to check for redundancies SD
Canonic treewidth t parses
It is known that b oth free and ro oted tree generation can b e done e ciently in
constant amortized time with one of the algorithms given in Wil WROM
BH To algebraically represent a graph of b ounded treewidth i e a generalized
b ounded width tree we mo del its underlying tree decomp osition structure as an
equivalent tree of maximum degree three using our treewidth op erator set see
t
Theorem Again since many tree decomp ositions of minimum width exist for a
given graph we strive for a canonic representation In the discussion given b elow we
incorp orate the additional binary op erator within the pathwidth lexicographical
canonic sc heme i e we expand the domain for the order
l
We view a treewidth t parse T as a ro oted parse tree where the current set of
active b oundary vertices are inferred from the op erator semantics
The rst simpli cation for our enumeration scheme is to restrict the t parse ar
guments for the op erator We simply require that no b oundary edges o ccur in
both t parses G and G b efore G G is enumerated Any edge that needs to b e
adjacenttotwoboundaryvertices is added with edge op erators after or ab ove the
circle plus
To compare two t parse s that have di erent underlying tree decomp osition trees
we use a ranking metho d similar to the one commonly used for ranking ro oted trees
BH The idea for our new linear order details to come shortly is to de ne another
CHAPTER BOUNDED COMBINATORIAL WIDTH
set of t parse equivalence classes with resp ect to each tree decomp osition structure
and then linearly order these classes
De nition Let v G denote the number of vertex op erators in a t parse G A
signature s G forat parse G is an integer sequence de ned recursively with resp ect
to the ro ot s op erator typ e
If G G ij thens G s G
n
If G G i then s G v G s G
If G G G where without loss of generality s G s G then s G
s G s G
A signature s is less than a signature s if js j js j or js j js j and s s in
lexicographic order
For anytwo t parses G and G that have the same signature let B B and
B B bethepathwidth t parse branches of G and G resp ectively obtained
from a p ost order traversal of the structural tree These vertex and edge op erators
are sequenced from leaves to ro ot Two t parses G and G not necessarily free
with B b oundary isomorphic can b e compared lexicographically by comparing B
i i
starting at i and increasing i until a di erence is found
Example A treewidth parse is shown b elowforS K a sub divided K
This pathwidth one obstruction is also shown in Figure a
n n n n n
A A A A A
With the far right edge op erator b eing the ro ot the signature of this b oundaried
graph is
CHAPTER BOUNDED COMBINATORIAL WIDTH
De nition A treewidth t parse G is structural ly canonic if it has the smallest
signature over all t parse representations of graphs free b oundary xed b oundary
isomorphic to G In addition the t parse G is treewidth canonic if it is the lexi
cographic minimum t parse over all structurally canonic representations within the
free b oundary xed b oundary isomorphic equivalence class We call these orderings
of treewidth t parses the free b oundary or xed b oundary lex rank order
It is understo o d from the context whether we are talking ab out the free or xed
b oundary cases with the latter case b eing more common
Lemma If a graph G of treewidth t also has pathwidth t then the structurally
canonic and treewidth canonic t parse for G has no op erators
Pro of This follows from the fact that a t parse with more op erators also has more
vertex op erators the circle plus op erator absorbs t vertices Since the length of a
signature for a t parse equals the number of vertex op erators any signature without
op erators i e a pathwidth t parse is less in the treewidth t parse comparison
order than any non trivial treewidth t parse
The ab ove lemma allows us to do computations enumerations for pathwidth
t parses and then reuse if using the lex canonic pathwidth scheme any partial results
for these t parses when computing within the treewidth t parse domain For example
for our obstruction set searches wewould keep any pro ofs of graph minimalityor
nonminimality from a searchthatwas restricted to pathwidth t parse s
Our treewidth analog to Lemma s pre x of canonic is canonic is given b elow
Lemma For the xed b oundary case any ro oted induced subtree S of a treewidth
canonic t parse T is also treewidth canonic
Pro of By induction on the numb er of op erators it su ces to lo ok at pre xes with
one less op erator Let g denote the last op erator of T Thevalidity of this statement
n
n
jk follows from the left asso ciative for the pathwidth op erators g i or g
n n
semantic interpretation of these unary op erators That is if there exists a pre x S
with a more canonic parse then applying the unary op erator g to S contradicts n
CHAPTER BOUNDED COMBINATORIAL WIDTH
T b eing treewidth canonic For the treewidth op erator case g assume T
n
G G is a treewidth canonic t parse By de nition of treewidth canonic we can also
assume G G in lex rank order If there exists a t parse H that is xed b oundary
isomorphic to G that has a smaller rank then the xed b oundary isomorphic t parse
T H G contradicts the fact that T is treewidth canonic This contradiction
can take place either in or outside T s structurally canonic equivalence class That
is if s H s G then we can nd a b etter structural equivalence class for T The
same argument holds for the child parse G replaced with G
As a consequence of the ab ove lemma there is a simple enumeration scheme
as was given in Section for the pathwidth t parses for generating all xed
b oundaried t parses of treewidth at most t Note that lik e our free b oundary path
width case extending a canonic xed b oundaried t parse G may yield a non canonic
t parse G Z butany pre x subtree of a xed b oundary canonic H is canonic An
example of the former problem is given b elow
G Z H
n n n n n n n n n n
The subtree prop erty of Lemma do es not hold for the free b oundary treewidth
case since the semantics of the binary op erator require xed b oundary vertices For
example consider the free b oundary canonic parse A B where A is taken from
n n n n
Example and B Here b oth subtrees A and B are
free b oundary isomorphic but B is not treewidth canonic for the free b oundary case
We end this section with an interesting op en computational problem regarding
how linear a tree decomp osition can b e for a given graph The structurally canonic
t parses in some sense have the simplest underlying treewidth structure for t b oundaried graphs see pro of of Lemma
CHAPTER BOUNDED COMBINATORIAL WIDTH
Problem Closeness to Pathwidth t
Input A graph G of treewidth t
Question How close is G to pathwidth t based on howmany op erators are needed
for a t parse representation That is howmany degree three vertices of T are needed
over all smo oth tree decomp ositions T fX j i T g of width t for G
i
a b
Figure The obstructions to a pathwidth and b treewidth Figure The minor order obstructions to treewidth
Chapter
Graph Minors and
Well Quasi Orders
In this chapter we formally intro duce the notion of minor order forbidden graphs
whichwehave b een calling obstructions We also explain why for many classes
of graphs i e the minor order lower ideals there exists a nite numb er of these
family characterizing graphs Lastly we de ne a closely related partial order for k
b oundaried graphs This order is the mathematical foundation for our obstruction
set search theory that is presented in Chapter
Preliminaries
We b egin with a review of a few de nitions from set theory that are signi cant for
characterizations of graph families by obstruction sets
De nition A quasi order preorder is a re exive and transitive binary relation
A partial order is an anti symmetric quasi order A partially ordered set is called a
poset
A simple example of a quasi order of graphs that is not a partial order is the
q
order de ned by comparing the number of vertices of the graphs i e G H if and
q
only if jGj jH j
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
De nition A set S with a quasi order is a wel l quasi order if for any count
able sequence s s ofmemb ers of S there are indices i j such that
s s
i j
Recall that a subgraph H of a graph G is a subset of the vertices and edges of
G such that whenever a vertex v V G n V H thenanyedgeof E G incidentto
v is not in E H We combinatorially contract an edge e u v inG to form a
edge contracted graph H G e by rst deleting e and then replacing vertices u and
v with a new vertex w suchthatan y edge incidentto u or v is now incidentto w
Also recall the following graph order that is of sp ecial interest to this dissertation
De nition For two graphs G and H the graph H is a minor of the graph G
written H G if a graph isomorphic to H can b e obtained from G by taking a
m
subgraph and then contracting p ossibly zero edges A minor operation is anyedge
deletion edge contraction or isolated vertex deletion
Observation The minor order relation de nes a partial order on the family
m
of nite simple graphs
Figure shows a Hasse diagram part of a minor order p oset for all edge
contractions of a graph with vertices
De nition A lower ideal I of a quasi order S is a subset I S such
that for any x I if y x then y I
For a sp ecial case wealsosay that a family F of graphs within the set of all
simple graphs is closed under the minor order if whenev er G is in F and H is a
minor of G H G together imply that H is also in F This equivalenttosaying
m
that F isalower ideal For minor order lower ideals the forbidden graphs are
m
de ned as follows
De nition The obstruction set O F for a minor closed family F of graphs is
F Sp eci cally G the set of minimal graphs in the minor order in the complement
F i e G F and for is an obstruction for F i e G O F if and only if G is in
any prop er minor H of G H is in F i e H F
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
a
b
c d
e
a b
c d
d e
Figure Illustrating an edge contraction p oset
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
For example the obstructions for the family of planar graphs are K and K
since any graph that is not planar must contain one of these two as a minor i e these
two obstructions are the minimal non planar graphs within the minor order p oset of
all graphs In general any graph in the complementofa lower ideal F must contain
at least one obstruction as a minor
The Graph Minor Theorem
How are obstruction sets used to characterize families of graphs that are closed under
the minor order It is not obvious that the set of obstructions for a minor order lower
ideal must b e nite The recently celebrated Graph Minor Theorem stated b elow
by Rob ertson and Seymour guarantees a nite characterization for all minor order
lower ideals RS b This statementwas known as Wagner s Conjecture for a long
time Wag RSc
Theorem Rob ertson Seymour The minor order is a well partial order
Corollary There are only a nite numb er of minor order minimal graphs in
any set of graphs In particular any minor order lower ideal of graphs has a nite
numb er of obstructions
Pro of Let S G G beanycountable sequence of distinct minor order min
imal graphs Since the minor order is a well partial order there must b e an index
i j suchthatG G However this contradicts the fact that G is minimal So
i m j j
there must b e a nite numb er of minor order minimal graphs
The bad news regarding this graph theoretical breakthrough is that Rob ertson
and Seymour s pro of is nonconstructive For more information see FRS This
means that for any set of graphs we only know that the set of minor order minimal
graphs is nite but currently have no general metho d for obtaining it To tackle this
predicament for our obstruction set computations we use the following variation of
the minor order The elements of this partial order are k b oundaried graphs
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
De nition Let G be a k b oundaried graph A k b oundaried graph H is a
minor boundaried minor of G denoted H G ifH is a minor of G such that
m
no b oundary vertices of G are deleted by the minor op erations and the b oundary
vertices of H are the same as the b oundary vertices of G
De nition Let G be a k b oundaried graph A k b oundaried graph H is a one
step minor of G if H is obtained from G by a single minor op eration one isolated
vertex deletion one edge deletion or one edge contraction and H G
m
Lemma The minor order on k b oundaried graphs is a partial order
m
Pro of Assume K H and H G for the k b oundaried graphs K H and G
m m
If we do the one step minor op erations used to show K H after the one step
m
minor op erations used to showthat H G we see that transitivity holds i e
m
K G Also since each one step minor of a graph reduces either the number
m
of edges or the number of vertices the minor order is anti symmetric
Lemma If a family F of graphs is a lower ideal under the minor order then
m
the family F f G S j G F and S is a k simplex of Gg of k b oundaried graphs
is a lower ideal in the minor order
m
Pro of This follows from the fact that the numb er of graph reducing minor op era
tions for the minor order is a subset of the minor order op erations
For our obstruction set searches concerning this minor order the width of each
k b oundaried graph is b ounded by a xed pathwidth or treewidth Sp eci cally we
htothesetof t parses t k restrict the searc
Lemma If a family F of graphs is a lower ideal under the minor order then the
family F consisting of F restricted to t parses is a lower ideal under the minor
order
m
Pro of Follows from Lemma and the next result Lemma which shows that
the set of t parses is closed under the minor order
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
The next two results showthatevery minor of a t parse has a representation as
a t parse Weknow that the family of b ounded pathwidth graphs is closed under the
minor order However this fact do es not guarantee that the b oundary vertices are
preserved by the minor op erations We actually implemented a di erent but similar
minor algorithm from the one used in the pro of b elow
Lemma Let G be a pathwidth t parse and supp ose H is a minor of G Then
H has a representation as a pathwidth t parse
Pro of Assume G g g g isthepathwidth t parse Weshowhow to con
n
struct a t parse for any one step minor of G The pathwidth t parse for any minor
H of G can then b e constructed by rep eating these one step minor pro cedures
Delete an isolated non b oundary vertex
n n
If op erator g u i nis b oth a non b oundary vertex i e g u for
i j
uv for u v and i k j some i j n and an isolated vertex i e g
k
then g g g g g represents a deleted isolated vertex minor
i i n
G nfug Note that we could have just as easily deleted g
j
Delete an edge
If g uv i n is any edge op erator then g g g g g
i i i n
represents a deleted edge minor G nf u v g
Contract a non b oundary edge
uv i nis any edge op erator such that for i c n If g
i
n n
g u or g v then do the following steps to create a contracted edge
c c
minor G u v First without loss of generality G is represented in one of the
following twoforms
n n n
g g u g v g uv g v g
a b i c n
or
n n n
g g u g v g uv g u g
a b i c n
n n
where g u for a j c and g v for b j c Note that the vertex
j j
n
op erator v may o ccur b etween g and g
a b
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
For the rst form use the t parse
g g g g g g g g g
a b c n
b i i c
where for b k c
ux if g vx and x v
k
g
k
g otherwise
k
to represent the minor G g
i
For the second form rst create from G a t parse g g g g g
c n
c
where for k c
ux if g vx and x fu v g
k
vx if g ux and x fu v g
k
n n
g
u if g v
k
k
n n
v if g u
k
g otherwise
k
to guarantee that the b oundary will b e preserved The new t parse satis es
so we can now use the previous construction The prepro cessing step is justi ed
n
since the op erator g u in the pre x graph G g g g tells us
c c c
which b oundary vertex to preserve that is the con tracted edge b etween vertices
represented by g and g b ecomes the activeboundaryvertex v in G u v
a b c
ij preventmulti edges from Recall that the semantics of the edge op erator
app earing
One can check that all three of these one step minor pro cedures will preserve G s
boundary
Observation Any minor of a treewidth t parse is a treewidth t parse
Pro of Deletion of an isolated non b oundary vertex and deletion of an edge is similar
n
to that in the pro of of Lemma An isolated vertex may originate from an i
app earing ab oveorbelow a circle plus in the t parse tree In this case we removethe
vertex op erator closest to the ro ot of the parse tree just ab ove the isolated vertex there
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
must b e one since we are trying to remove a non b oundary vertex This op eration
is correct b ecause the op erator merges together two previous isolated b oundary
vertices with the same lab el from di erent subtree parses For the edge deletion
case with resp ect to we also have to b e careful ab out two ij op erators b eing
used to represent the same edge However removing all instances is easy
Contracting a non b oundary edge is also p ossible by using a mo di ed version
of the pathwidth t parse contraction algorithm Let e u v b e an edge that we
want to contract For these twovertices u and v we remove the corresp onding vertex
op erator closer to the ro ot instead of the lower indexed one Again wemayhaveto
permute the b oundary to get the t parse branches to satisfy this happ ens when
n
twovertex op erators nearest the edge op erator are b oth v see pro of of Lemma
De nition A t parse b oundaried graph G is a obstruction boundaried ob
struction ofalower ideal F if every prop er minor of G is in F while G is not in
F
For any minor order lower ideal F and an obstruction O O F there exists a
obstruction G in such that O G trivially whenever t pw O In this case the
m
t
b oundary of G with t vertices can b e any set of vertices X from a smo oth path
t t t
decomp osition X X X of width t for O LetB f G S j i r g
r t
i i
denote the set of obstructions of width t for F Ift is the pathwidth of the largest
obstruction for a lower ideal F then
j
A
G O F
i
i r j t
j
The previous discussion also holds for obstructions of b ounded treewidth That is
for any smo oth tree decomp osition T fT g ofa graph G of width t there exists a
x
treewidth t parse with the b oundary of G s representation b eing the vertices of anyset
T of T So the set of underlying b oundaried obstructions for b ounded treewidth
x
is a sup erset of the non b oundary obstructions
Lemma If F is a lower ideal in the minor order then there are a nite number
of obstructions in the minor order for F
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
Pro of It su ces to showthatany minor order obstruction O O F is b elowat
most a nite number of obstructions in the minor order Let O B for some
m
obstruction B There is a sequence of length at most t of edge contractions b etween
two b oundary vertices and or isolated b oundary vertex deletions to pro duce O from
B If any other one step minor op erations were p ossible then B would not b e a
obstruction For example deleting any edge can b e done rst whichisavalid
minor op eration showing that B has a minor not in F Therefore any obstruction
ab ove O can have at most jO j t vertices
Even though Lemma guarantees that there are only a nite numb er of b ound
aried obstructions for any minor order lower ideal wehave the following analog to
the Graph Minor Theorem for t b oundaried graphs whichwas rst cited without
pro of and used in FL a The follo wing simple pro of uses Rob ertson and Seymour s
more recent but less known result that the set of nitely edge colored graphs is also
well quasi ordered under the minor order denoted by The pro of should app ear
cm
in RSf
Theorem Fellows Langston The minor order is a well partial order on t
b oundaried graphs and t parse s
Pro of Given anycountable sequence G G of t b oundaried graphs weneedto
show that there exist indices i j suchthat G G Todothiswe rst de ne
i m j
a bijection as illustrated in Figure from t b oundaried graphs to edge colored
graphs using t colors For eachboundaryvertex lab eled k the function adds
a single k colored p endent edge while removing the b oundary lab el The original
noncolored edges in the domain graph of retain the blank color in the image graph
Thus this map pro duces a total t colors in the image
Since the range of is a well partial order we consider the following sequence of
t edge colored graphs G G Here there must exist a G G
i cm j
for i j Toprove our theorem we showthat
H H K K H K
cm m
Let M m m m b e the sequence of one step minor op erations that shows
r
H K Since b oth H and K have exactly one edge of color through t none cm
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
of these colored edges were contracted or deleted within M Also since none of the
colored edges t in H are adjacent there are no edge contractions b etween
any edge u v E K where u and v are b oth incident to colored edges of K
Thus M constitutes a valid sequence of b oundary minor op erations on the underlying
b oundaried graph K K showing that H K
m
Figure A map from b oundaried graphs to edge colored graphs
Other Graph Partial Orders
This chapter s nal section summerizes some other partial orders for graphs that
app ear in the eld of graph theory We remind the reader that the obstruction set
characterizations presented in later chapters of this dissertation are all based on the
minor order
Perhaps the most natural partial order for graphs is the subgraph partial order
Here a graph H is b elow another graph G often denoted H G if it is a subgraph
A slightvariation of this order is call the vertex induced subgraph partial order Here
a graph H is an induced subgraph of a graph G often denoted H G if there exists
a subset V of V G suchthat H G n V We presented in Chapter the induced
subgraph obstructions known as asteroidal triples that preventchordal graphs from
being interval graphs
It is easy to see that the subgraph and induced subgraph partial orders are not
well partial orders since for example the set of cycles fC j i g is an in nite
i
an ti chain i e a countable set of incomparable graphs
There are twowell known intermediate partial orders of graphs that fall b etween
the structured minor order and the simple subgraph order The rst order is the
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
topological order We de ne the top ological partial order as a restricted minor order
where only subgraphs and edges incident to a degree twovertex maybecontracted
Other pap ers alternately de ne the top ological order on graphs based on the fol
lowing relation e g see Wil Here a graph G top ologically contains H often
denoted H G if some subgraph of G is isomorphic to a sub division of H where to
t
sub divide a graph is to replace its edges by paths that are vertex disjoint except at
endp oints The second intermediate order is the induced minor order Here a graph
H is an induced minor of a graph G often denoted H G ifH can b e obtained
im
bycontracting edges from a induced subgraph of G
For the top ological order Kruskal proved the following well known result Kru
Theorem Kruskal s Tree Theorem The top ological order is a well quasi order
for the set of nite trees
A stronger result stating that all nite ro oted trees are well quasi ordered for the
top ological order with a simple pro of is due to Nash Williams NW Since the
minor order is a stronger relation Kruskal s Tree Theorem immediately implies that
the minor order is a well partial order for trees
The immersion order on graphs is de ned by three typ es of graph reductions
i
deleting vertices deleting edges or lifting edges Tw o adjacent edges a b and b c
are lifted from a graph if they are replaced with a single edge a c See Figure a
In addition to the Graph Minor Theorem Rob ertson and Seymour have recently
proven that the immersion order on graphs is a well partial order The pro of of this
fact will app ear in RSf of the graph minor series
a b Figure Illustrating the a lift and b fracture graph op erations
CHAPTER GRAPH MINORS AND WELL QUASI ORDERS
A related partial order termed the weak immersion order is de ned by allow
wi
inggraphstoberelatedby rep eated uses of these four graph op erations delete an
isolated vertex delete an edge remove a sub division or fracture avertex A vertex
v is fractured by replacing it with two new vertices v and v and partitioning the
edges incidenton v into two classes making these incident resp ectively on v and v
Fel See Figure b
Makedon and Sudb orough showed that the family of graphs with cutwidth at
most is characterized by a set of ve forbidden weak immersion order multi graphs
MS these graphs are displa yed in Figure b elow The family k Cutwidth
graphs with cutwidth at most k is also a lower ideal for the immersion order e g
see Ram
Figure The weak immersion order obstruction set for Cutwidth
Chapter
Finding Forbidden Minors
We search for obstructions within the set of graphs of boundedpathwidth or bounded
treewidth Wenow describ e howtodothise ciently with our algebraic t parse
representation For ease of exp osition throughout this chapter we limit ourselves to
b ounded pathwidth graphs in the obstruction set searchtheory and only p oint out
further information regarding a b ounded treewidth search as needed
Key t parse Prop erties
Recall that our concatenation partial order over t parse s is based on strings or trees
of t parse op erators This ordering of graphs is used during our enumeration of
the graphs of b ounded combinatorial width As explained b elow this enumeration
pro cess can b e restricted to a nite search space consisting of the minimal graphs
in the b oundary minor order intersected with any nite index family lower ideal
congruence
As wesaw in Chapter b oth the k Pathwidth and k Treewidth families
of graphs are lower ideals in the minor order so a minor H of a t parse G is
representable as a t parse One should keep in mind that these minor order algorithms
can actually op erate on t parses directly bypassing an y unnecessary conversion to and
from the standard graph representations
CHAPTER FINDING FORBIDDEN MINORS
The following sequence of de nitions and results form our theoretical basis for
computing minor order obstruction sets We rst recall a de nition for the more
familiar and general setting of k b oundaried graphs e g see FL a
De nition Let F b e a xed graph family and let G and H be k b oundaried
graphs Wesaythat G and H are congruent with resp ect to the canonical congruence
if for each k b oundaried graph Z
F
G Z F H Z F
A k b oundaried graph Z is a distinguisher for G H if G Z F and H Z F
F
or vice versa
For our setting of graphs of b ounded combinatorial width t parse s wehavethe
following analogous de nition
De nition Let F b e a xed graph family and let G and H be t parse s Wesay
that G and H are F congruent also written G H ifforeach extension Z
F
t
G Z F H Z F
If G is not congruenttoH thenwesay G is distinguished from H by Z and Z is
a distinguisher for G and H Otherwise G and H agree on Z
We call the ab ove family congruence the bounded width canonical congruence
since it resembles the general canonical congruence for F on t b oundaried graphs
However we also use the phrase canonical congruence lo osely to mean either case
dep ending on the context The only di erence here is that we restrict each Z inthe
latter case to b e a parse string representing a graph of b ounded combinatorial width
oncatenation op erator Recall that the op erator is called the t parse c
The next de nition provides a means for distinguishing graphs in di erent equiv
alence classes of the canonical congruence
F
De nition A set T or a set T of b oundaried graphs is a testset if
t
G H implies there exists a Z T that distinguishes G and H with resp ect to F F
CHAPTER FINDING FORBIDDEN MINORS
As shown later a testset is only useful for nding obstruction sets if it has
nite cardinality Weareinterested in canonical congruences for our obstruction set
computations b ecause of the following consequence of the GMT see AF and
Cou a Cou b
Observation The canonical congruence for any minor order lower ideal F is
of nite index for the set of t parses
Pro of This result follows by using the following two facts the GMT implies that
the family F has a nite numb er of obstructions and there exists a nite state
algorithm for each minor containment problem for input graphs of b ounded treewidth
by Courcelle and others The second fact means that for the family of graphs that
excludes a single obstruction there is a nite state automaton that accepts t parses
in that family Since regular languages are closed under intersections the b ounded
width canonical congruence is of nite index See Section and Chapter
F
for further information
The ab ove observation can b e improved to show that the canonical congruence
F
is of nite index for the set of k b oundaried graphs This is b ecause all obstructions
of the lower ideal F have b ounded treewidth A nite testset for F is constructed by
taking a nite union of nite testsets of b oundary size k for eachwidth t up to the
treewidth b ound
De nition A t parse G is nonminimal if G has a prop er minor H such that
G H Otherwise wesay G is minimal A t parse G is a obstruction if G is
F
minimal and is not a member of F
In general if a graph family F is a minor order lower ideal and a t parse G is
minimal then for each minor H of G there exists an extension Z such that
G Z F and
H Z F
That is there exists a distinguisher in this direction for each p ossible minor H of
the t parse G
CHAPTER FINDING FORBIDDEN MINORS
The obstruction set O for a family F is obtainable from the b oundaried obstruc
F
tion set O set of obstructions bycontracting b oundary edges or deleting isolated
F
b oundary vertices whenever the search space of width is large enough We will
use the symbol O or O F to also denote O since the typ e of the obstructions
F
F
will b e clear from the context
In our search for O wemust prove that each t parse generated is minimal
F
or nonminimal The following two results drastically reduce the computation time
required to determine these pro ofs
Lemma A t parse G is minimal if and only if G is distinguished from
each one step minor of G Or equivalently G is nonminimal if and only if
G is F congruenttoaone step minor
Pro of Weprove the second statement Let G b e nonminimal and supp ose there
exists two minors K and H of G suchthat K H and K G It is su cientto
m F
show H G
F
For any extension Z ifG Z F then H Z F since H Z G Z and
m
t
F is a minor lower ideal NowletZ be any extension such that G Z F Since
Z wealsohave H Z F K G wehave K Z F And since K Z H
F m
Therefore G is F congruentto H
Lemma Pre x Lemma If G g g g is a minimal t parse
n n
then any pre x t parse G m n is also minimal
m
Pro of Assume G is nonminimal It su ces to showthatany extension of G is
n n
nonminimal Without loss of generality let H b e a one step minor of G such that
n
for any Z
t
G Z F H Z F
n
Let g and G G g Because of the semantics of the op erator set
n t n n n
H H g is a one step minor of G such that for any Z
t n n
t
Z F H g Z H Z F G Z G g
n n n n
Thus any extension of G is nonminimal n
CHAPTER FINDING FORBIDDEN MINORS
The ab ovetwolemmas also hold when the circle plus op erator is included in
i e treewidth t parses For illustration consider the Pre x Lemma If G is a
t
nonminimal t parse with an F congruentminorG andZ is any t parse then G Z
is an F congruent minor of a nonminimal G Z where we use the prime symbol to
denote the corresp onding minor op eration done to the G part of G Z Theawkward
notation is needed since G Z mayequalG Z when common b oundary edges exist
in b oth G and Z
A Simple Pro cedure for Finding Obstructions
The Pre x Lemma implies that every minimal t parse is obtainable by extending
some minimal t parse this pro vides a nite tree structure for the search space This
guarantee of termination is proven at the end of this section Also since a t
b oundaried graph mayhavemany t parse representations we can further reduce the
size of the searchtreeby enforcing a canonical structure on the t parses enumerated
To do this we ensure that every pre x of every canonic t parse is also canonic recall
Chapter Combining these two ideas we can search for minor order obstructions
by using the following simple pro cedure
n n n
Start with the empty t parse t as the initial minimal
t parse Since there are no one step b oundary minors this t parse is vacuously
minimal Add the t parse to a grow set Gr ow
If there exists a t parse G in the grow set Gr ow then enumerate all one op erator
extended t parses E E E of G that are canonic and relevant
r j j
t
Otherwise stop
For each t parse E i r eitherprove it minimal or nonminimal
i
G from Gr ow and add each minimal t parse E i r toGr ow Remove
i
Return to step
See the sample search tree given in Figure We used the term relevant ab ove
to mean that wemight only b e interested in obstructions for a restricted subset
CHAPTER FINDING FORBIDDEN MINORS
of t parses For example three p opular classes of relevant t parses in addition to
just the free b oundary isomorphic graphs are connected graphs b ounded degree
graphs and planar graphs Each class is intersected with the graphs of b ounded
combinatorial width t
This search pro cedure bypasses many graphs in a lower ideal F usually of in nite
cardinality when branching out towards the nite numb er of obstructions minimal
leaves of the search tree Recall that the set of minimal t parses contains the set of
b oundaried obstructions of width t i e the non family minimal t parses are precisely
the b oundaried obstructions
Step intheabove pro cedure is the hardest to compute and we will dedicate
most of the remainder part of this chapter to the issue of proving t parses minimal or
nonminimal Also notice what happ ens when r at step For this situation we
do not expand the search tree at a minimal t parse this is wh y there is a white circle
leaf in our sample search tree of Figure For example for our canonic t parse
enumeration scheme it is easy to see that the clique K is a canonic dead end for
t
t
Now that wehave presented our basic metho d for nding obstruction sets we use
the Pre x Lemma as promised ab ove to prove that the pro cedure will terminate
Theorem F oralower ideal F with an available decision algorithm for com
puting i e for proving t parse minimality there exists a terminating algorithm
F
e g the one given ab ove to compute O F for eachpathwidth t
Pro of It su ces to provethatwe will generate only a nite number m of minimal
t parses The Pre x Lemma allows us to prune at eachofatmost O m nonmin
imal t parses Our search tree has b ounded out degree consisting of the number of
op erators in i e a function of the width t Recall that has nite index If
t F
there is not a nite number m of minimal t parse s then there must b e an in nite set
S of distinct t parses in some equivalence class But since the b oundary minor
F
order is a well partial order there must exist two t parses G and G in S such that
G G This implies that the t parse G has G as a congruent minor This
m
contradicts the fact that G was minimal Therefore our search will stop shortly after
all of the nite numb er of minimal and canonic t parses havebeenenumerated
CHAPTER FINDING FORBIDDEN MINORS
Nonminimal in F leaf
Minimal t parse in F
Nonminimal not in F leaf
Minimal t parse not in F
obstruction leaf
Figure A typical t parse search tree each edge denotes one op erator
CHAPTER FINDING FORBIDDEN MINORS
We p oint out that for treewidth t parse searches wealsohave a easy pro of of
termination However in this case the out degree of the search tree changes for each
minimal t parse G This is b ecause concatenation also includes the use of the
op erator Here G G is enumerated for every other treewidth canonic op erand G
such that G is less than G in the lex rank order see De nition and Section
Proving t parses Minimal or Nonminimal
We currently use the following four techniques to prove that a t parse in the search
tree is minimal or nonminimal
Direct nonminimalitychecks
Dynamic programming congruences
Random extension searches
Canonical congruence testsets
These steps are listed in the order that they are attempted if available if one
succeeds the remainder do not need to b e p erformed The rst three of these may
not succeed though the fourth metho d always will
Note that a b oundaried obstruction G is the only typ e of minimal t parse that
has an empty extension Z that distinguishes eachminorG G from G In
m
these trivial cases w e do not need to use any of the ab ove four pro of metho ds
By default the third metho d of doing random extension searches is always avail
able whenever wehave a decision algorithm for a lower ideal F All of the ab ove
techniques are optional in the sense that wemay only need or want to use just one
metho d Wenow list three common situations as examples If wehave the canonical
congruence for F implemented as a dynamic program then the second metho d
F
is su cient e g see our k Vertex Cover characterizations If wehave an algo
rithm that detects al l nonminimal graphs then the rst metho d is su cient e g see
CHAPTER FINDING FORBIDDEN MINORS
our k Feedback Edge Set characterizations Lastly ifwe are only interested in
computing a partial set of obstructions with a go o d chance of nding the complete
set we can use only the third metho d
Atypical obstruction set search will use all four pro of techniques which are
explained in more detail in the following subsections
Direct pro ofs of nonminimali ty
There exist easily observable prop erties of t parses called pretests that imply non
minimality For any t b oundaried graph t parse wehave to assume that the
boundary is completely connected or has the p otential to b e This is b ecause we
do not want to eliminate out any pre x that may b e extended to an obstruction
For an example of a pretest consider the k Feedback Vertex Set family of
Chapter The existence of a degree one vertex in the interior is a trivial example
of such a nonminimal prop erty Infactmany pretests that we use are easy to prove
To illustrate this p oint further wehave the following pretest
Lemma Any graph with resp ect to the k Feedback Vertex Set family that
contains a cycle with two degree twovertices is nonminimal
Pro of Weknow that if a graph G has adjacent degree twovertices then it is nonmin
imal except for the C cases in fact this is another simple pretest This is b ecause
the minor G e created by contracting the edge e between these degree twovertices
still requires a minimum feedbackvertex set of the same size Alternatively stated
FV S G FV S G e So we just need to consider a graph G with a cycle that
lo oks like
a a
u v u v
or
b b
For either case we claim that the edge contracted minor G G a u is congruent
to G i e a witness to nonminimality This is easily seen since any feedbackvertex
CHAPTER FINDING FORBIDDEN MINORS
set V of G or any extension of G that contains vertex u or v can b e replaced with
V V nfug fbg or V V nfv g fbg This set V is also a feedbackvertex
set for G Likewise if W is a feedbackvertex set for G then W is also a feedback
vertex set for G
If there exists an e cient algorithm for a pretest then wemay opt to test it on
eachenumerated t parse b efore invoking our canonicity algorithm The reason is that
if we get an easy nonminimal pro of for a t parse then we do not care whether it is
canonic or not i e it do es not matter whether wehave found a nonminimal dead
end in our search tree or a redundant representation Our currentsystemdoesthe
canonicity testing in tw o parts
Check for the easy non canonic prop erties using the results of Chapter i e
one can think of this step as a non canonic pretest
Use a slower algorithm e g a brute force back tracking metho d that tests for
t parse canonicity
For each newly enumerated t parse we then try all the fast nonminimality pretests
between these two canonic stages
For our t parse enumeration scheme a related pretest typ e pruning is presented
below in Section Here for certain lower ideals it is p ossible to prune the search
tree at the disconnected t parses regardless of their minimal nonminimal status
Pro ofs based on a dynamic programming algorithm
We can sometimes use a dynamic programming algorithm A as a nite re nement
of the canonical congruence for a xed width We can view each of these
F
algorithms as a nite state linear tree automaton for F bysimulation This means
that there is a mapping from the dynamic programming states of A equivalence
classes of onto the equivalence classes of The input to a b ounded width t
F
t
is any t parse over the alphab et These algorithms always nite state algorithm A
t
run in linear time where the size of the input is the number of t parse op erators but
CHAPTER FINDING FORBIDDEN MINORS
may contain large hidden constants that are a function of b oth t width and k
where k is a p ossible parameterized integer i e k is some index in a series of families
fF F F g
k
t
For such a nite state algorithm A for F ifbotha t parse G and a one step
minor G of G end up in the same computation state then G G This is b ecause
F
G G G G Then by de nition the t parse G is nonminimal However if
F
G and G have distinct nal states no conclusion can b e reached If every one step
minor G is not in the same nal state as G then we usually pro ceed with the next
t
step given in Section the exception b eing when the states of A corresp ond
in a one to one fashion with the equivalence classes of the canonical congruence
F
t
In this latter case we call A a minimal nite state algorithm for F for width t
If we are fortunate to have a minimal nite state algorithm then we can also prove
minimality That is we can use the nite state algorithm to provide b oth minimal
and nonminimal pro ofs This is b ecause distinct nal states imply the existence of an
extension that distinguishes the two states and their t parse representatives More
t
formally a nite state algorithm A is minimal with resp ect to a lower ideal F if
for anytwo t parses G and H wehave or sp eci cally
F
for all Z G Z F H Z F G H
t
t
where G H means that G and H end up in the same state of A An example of such
a minimal nite state algorithm is presented in Chapter for the k Vertex Cover
graph families
Now consider the rare case that a nite index congruence implemented as some
typ e of dynamic program requires substantial computational e ort to up date a
t parse s state Here we are considering the time to app end a single op erator to
the t parse Wehave some p ossible remedies when using this typ e of dynamic pro
gram as a congruence for our obstruction set computations
We can keep a state lo okup table for eachcanonict parse Thus during the
enumeration of t parse s we do one dynamic programming step with a single
op erator applied to a previous state of a pre x t parse This previous state
is obtained from the lo okup table instead of recomputing several states for the
CHAPTER FINDING FORBIDDEN MINORS
length of a t parse One unfortunate consequence of this is that wemayhave
to determine states for some nonminimal t parses These graphs o ccur from the
set of one step minors of a given t parse For these cases we up date the states
for several op erators starting from a minimal t parse that is a pre x
t
From the dynamic programming congruence A for a lower ideal F wecan
initially build a deterministic nite state automaton M that accepts t parses in
F This automaton provides fast single step state transitions Alternatively
we can use a canonical congruence implemented as a testset to automatically
F
create M See b elow for more information and Sections for howto
build these automata The only aw with either of these constructions is that
the generated automaton may b e to o large to store in memory even after b eing
minimized
We end this subsection with a short case study Weshowhow to construct
dynamic programming congruences for the k Edge Bounded IndSet lower ideals
Recall that memb ers of these families have indep endence size k The reader
is invited to skip to the next t parse minimality nonminimality pro of metho d Sec
tion if these details are not of interest
We rst consider a dynamic program that determines the indep endence of a path
width t parse This algorithm I d p is presented in Figure and proven correct
below in Lemma For the related k Edge Bounded IndSet family a b oundaried
graph congruence can b e sp eci ed as follows For a t parse G and each subset S of
the b oundary de ne the following substates whenever p ossible
I ndS et S max jI j k I S and I is an indep endent set
G
Two t parses G and H are in the same equivalence class if
min jE G j k min jE H j k and
I ndS et S I ndS et S for all S
G H
entation we use a sp ecial symbol void to For our dynamic programming implem
denote that there is no indep endent set containing some S That is the following
CHAPTER FINDING FORBIDDEN MINORS
Initial Conditions
n n
For the pre x G t ofat parse G g g g set
t n n
I ndS et S jS j for each S
Lo op
For each op erator g k t n do the following over all S
k
n
Case For new replacement b oundary vertex g i and i S
k
I ndS et S max I ndS et S I ndS et S nfig
n
and i S Case For new b oundary vertex g i
k
I ndS et S max IndSet S IndSet S fig
Case For new b oundary edge g ij with i S and j S
k
IndSet S void
Case For new b oundary edge g ij with i S or j S
k
I ndS et S I ndS et S
On Exit
Return maximum value of any IndSet S over all S
Figure A general indep endence algorithm I d p for pathwidth t parses
CHAPTER FINDING FORBIDDEN MINORS
simple alteration to the ab ove substates is used
void if S is not an indep endent set of G else
I ndS et S
G
max jI j k I S and I is an indep endentset
Next we present a pro of that the in nite state dynamic program I d p correctly
up dates these I ndS et S substates in the t parse setting
Lemma Consider the indep endence algorithm I d p given in Figure For
any pathwidth t parse G and extension op erator g the IndSet S substates of
t
I d p are correctly up dated for G g
Pro of By de nition of I ndS et S for S the state IndSet S jS j for the
n n n
empty t parse G t The four dynamic programming cases handle
whether g is a vertex or edge op erator and the sp eci c index S Inthe
case analysis b elow recall that a witness I for I ndS et S is an indep endent set of the
t parse such that I S The notation IndSet S denotes the substate of G g
while I ndS et S denotes the previous state of G
n
Case g i where i S
Let I and I b e maximum indep endent sets for IndSet S and I ndS et S nfig such
i
that I S and I S nfig F or a new isolated b oundary vertex i we see that
i
IndSet S max IndSet S IndSet S nfig b ecause either one of the sets I or
I plus one additional vertex is a witness Toshow that equality holds we consider
i
the following two cases when I ndS et S is at least two greater than b oth I ndS et S
and I ndS et S nfig If I is a witness for IndSet S then consider I I nfi g where
n
i denotes the new b oundary vertex of G i First if I contains the b oundary
vertex i of G then I S of larger cardinality Second if I is a witness for I ndS et
do es not contain the b oundary vertex i of G then I is a witness for I ndS et S nfig of
larger cardinality Both these contradictions show that this dynamic step is correct
n
Case g i where i S
Now let I and I b e indep endent set witnesses for b oth I ndS et S and I ndS et S
i
fig resp ectively Sincei S both I and I are also maximal indep endent sets with
resp ect to IndSet S Thus IndSet S max IndSet S IndSet S fig Again
CHAPTER FINDING FORBIDDEN MINORS
as in case equalitymust hold otherwise we can contradict one of the previous
maximum values IndSet S or I ndS et S fig
Case g ij with i S and j S
There is no p ossible indep endent set when b oth vertices i and j are adjacent so
the assignment IndSet S void is correct In future steps we de ne max n void
n for anyinteger n that is void is less than anyinteger n in this linear order
ij with i S or j S Case g
Since a witness indep endent set I for IndSet S is also a witness for I ndS et S
and the indep endence of a graph do es not increase when edges are added no change
is needed for this situation
From the ab ove lemma it follows that we can determine in linear time the max
imum indep endent set of a t parse G by running the I d p algorithm The in
dep endence of the graph is then the largest IndSet S value To actually nd an
indep endent set of maximum cardinality a history of witness vertex sets i e one
indep endentset I for each I ndS et S is kept and up dated during the dynamic steps
For each xed parameter k itiseasytoconvert I d p into a nite state al
t
gorithm A that can determine if a t parse has an indep endent set of size greater
k
than k This is done by simply assigning k toany I ndS et S thatever exceeds
t
k Wedonotknow whether this nite state algorithm A is the smallest in
k
terms of the numb er of computation states required This implied graph family in
dep endence greater than k isnotalower ideal in the minor order However we
t
can adapt each nite state algorithm A into a dynamic programming congruence
k
for each k Edge Bounded IndSet lower ideal
Lemma Let F k Edge Bounded IndSet For each b oundary size t an
t
upp er b ound on the numb er of equivalence classes for is k
F
Pro of Consider the k Edge Bounded IndSet congruence given earlier For each
subset of the b oundary S I ndS et S is assigned either one of the integers f k g
t
or the sp ecial symbol void Since there are p ossible choices for S and each graph
CHAPTER FINDING FORBIDDEN MINORS
in the lower ideal can haveatmostk edges again an integer f k g the
b ound follows
We close this section bymentioning that the ab ove b ound is probably not very
strong for k Edge Bounded IndSet Ifweknow that a b oundaried graph has k
edges then we should b e able to restrict to the substates of the nite state indep en
t
dence algorithm for A
k k
Pro ofs obtained by a randomized search
We can provethata t parse G is minimal by nding for each one step minor G
of G a distinguisher Z that is a parse extension Z suchthatG Z is a graph in F
while G Z is not a graph in F Recall Lemma
In the exp erimental results describ ed in Part I I of this dissertation a large fraction
of the minimality pro ofs in the search trees were obtained by a randomized algorithm
This randomized algorithm consists of the following high level steps for a t parse G
Initialize L fG j G is a one step minor of Gg
Until L or until a time limit is exceeded do
Cho ose G L
Randomly generate a distinguisher Z for G and G
Pick a minor Z Z suchthatZ is a distinguisher for G and G
m
but for every prop er minor Z Z Z do es not distinguish G and G
m
Remove G and then any other G L suchthat G Z F
The goal of this algorithm is to nd a distinguisher for each one step minor In
ecanusea di erent distinguisher for each one step minor in a pro of of particular w
minimality Step cho oses a one step minor as a target Step whichisthe
most computationally intensive step may not succeed in the given time limit If it is
successful then steps and extract as much useful information as p ossible from
this distinguisher In practice we found that many one step minors of G could b e
distinguished from G by the same extension Z The minimization of Z in creating
CHAPTER FINDING FORBIDDEN MINORS
Z in step increases the chances that the extension distinguishes G from other
one step minors
Example Consider the family F Vertex Cover The gure b elow de
picts a t parse G for t along with two randomly generated extended graphs
G Z and G Z that provides a pro of of minimalityfor G
n n n n n
The t parse G K K
and Z The two pro of extensions are Z
G Z G Z
G
The compact visual representation of the one step minors consist of a b old edge is
an edge contracted minor a dashed edge is an edge deleted minor and a b old dashed
edge indicates two di erent minors each one obtained by an edge deletion or an edge
contraction The extension gure in the middle shows ve one step minors of G and
an extension Z that gives the desired minimality pro of for all those ve minors The
extension gure on the right nishes the t parse minimality pro of using Z for the
remaining single edge deleted minor
We end this subsection with two remarks concerning p ossible uses of random
distinguishers
In our exp erimental work we found that a careful examination of those cases
where a randomized search for a pro of of minimality failed and hence for which
this failure provides circumstantial evidence of nonminimality often provided
structural insights that could b e translated into new nonminimality pretests
CHAPTER FINDING FORBIDDEN MINORS
Often go o d candidates for distinguishers can b e obtained from partial sets of
computed obstructions Here a piece b oundaried subgraph of an obstruction
has minimality prop erties for the currentlower ideal Trivially the single edges
via a single edge op erator extension are pieces of obstructions Thus for any
interesting lower ideal we could start with these edge pieces even b efore we nd
our rst obstruction
Pro ofs based on a testset congruence
We can use a complete testset see De nition to determine if a t parse G is
distinguished from each of its one step minors recall Lemma If so the t parse
G is minimal by a testset pro of Alternatively a t parse G is nonminimal if and only
if it has a one step minor G suchthat G and G agree on every test Thus after a
testset run wehave a pro of that a graph is either minimal or nonminimal
It helps to have a testset of minimum size and a fast memb ership algorithm for
F Unfortunately itis NP hard to nd the minimum sized testset for the minimal
automaton or canonical congruence for F See Chapter for a pro of However
there exist heuristics that to help nd redundant tests in a testset For example
we can remove one test and see if the remaining tests de ne the same congruence
That is we can compare the minimal automata generated by these two testsets See
Section for instructions
All of our testsets that are presented in Part I I w ere proven complete for the set
of all b oundaried graphs It is easy to see that if two graphs are congruent for the set
of all t b oundaried graphs then they are also congruent for the set of t parse s
Wepoint out that testsets exist for other regular language graph families b e
sides the b ounded width minor order lower ideals The set F of t b oundaried
HC
graphs with a Hamiltonian cycle is a classic example In Figure we illustrate a
testset for F for graphs of b oundary size Recall that a graph G is Hamiltonian
HC
if there exist a cycle of length jGj
The pro of that these tests are su cientisdoneby a case analysis regarding how
a Hamiltonian cycle may pass across the b oundary We provide a simple pro of b elow
CHAPTER FINDING FORBIDDEN MINORS
Figure The set T of b oundaried graph tests for Hamiltonicity
HC
in order to illustrate how one generally constructs testsets for an arbitrary family F
In establishing most of our testsets for lower ideals we use this basic pro of technique
is a Hamiltonian cycle testset for the set of b oundaried Lemma The set T
HC
graphs
Pro of Let denote the canonical Hamiltonian cycle family congruence on graphs
HC
of b oundary size Recall that G H if and only if for every b oundaried graph
HC
extension Z
G Z F H Z F
HC HC
If G H then without loss of generality wehavesomeZ such that G Z F
HC HC
and H Z F Wejustneedtoshow that this distinguishing Z can b e replaced
HC
with one of the tests in T de ned in Figure Let C be any Hamiltonian
HC
also distinguishes G cycle of G Z Clearly the b oundaried graph Z Z C
and H This is b ecause non Hamiltonicity is closed under edge deletions note that
V Z V Z Now Z must consist of a set of connecting paths or unique cycle
between the b oundary vertices Again since non Hamiltonicity is closed under edge
CHAPTER FINDING FORBIDDEN MINORS
contractions b etween adjacent degree twovertices Z can b e replaced with one of the
reduced tests in T
HC
Making the Theory Practical
This section contains a selection of results that help sp eed up the pro cess of computing
obstruction sets
One computational improvement deals with determining whether a t parse G
is minimal or nonminimal when G is not in a lower ideal F By de nition of the
canonical congruence for F we see that G is minimal if and only if for every
F
minor H of G the minor H is in F Otherwise if H is not in F then H G
F
Thus minimality is easily determined bychecking that each one step minor of an
out of family t parse is in the lower ideal F
Pruning at disconnected t parses
Often for a lower ideal F we are only interested in computing the connected obstruc
tions since as explained in this section the disconnected ones are easy to derive
Belowwe indicate a general setting where we can prune the search tree whenever a
disconnected b oundaried graph is enumerated Here disconnected means that there
is a connected comp onent containing only non b oundary interior vertices
Let b e a function that maps graphs to non negativeintegers such that
For graphs G and G G G G G
For any minor H of G H G and
For any graph G there exists a minor H such that H G
The family of graphs F k fG j G k g k has an obstruction set
wer ideal from prop erty above in the minor order Also O F k since it is a lo
has nite cardinality b ecause of the Graph Minor Theorem A concrete example
is G g enus G where genus G denotes the smallest genus of all orientable
CHAPTER FINDING FORBIDDEN MINORS
surfaces on which G can b e emb edded prop erty follows from BHKY In fact
there are many examples of lower ideals that satisfy all three prop erties such as all
of the within k vertices families characterized in this dissertation
Observation If F is a lower ideal suchthat
G F and G F G G F
then the function
G min jV j G n V F where V V
can b e used to de ne the ab ove parameterized lower ideals F k k
Pro of Clearly G G G G since if G n V F and G n V F
then G G n V V F Likewise G G G G since if
G G n V F then G n V G V F and G n V G V F So
y holds for Prop erty follows from the fact that F k is de ned as a prop ert
within k vertices family of a lower ideal see FL b Prop erty follows from the
fact that if H is any minor obtained from G by deleting a single vertex then H
can b e at most one less than G
For example our k Vertex Cover and k Feedback Vertex Set parameter
ized lower ideals are de ned from the base lower ideals F fgraphs with no edgesg
VC
and F fgraphs with no cyclesg resp ectively Here the families F and F
FV S VC FV S
are closed under graph unions so Observation is valid
Theorem If G C C is an obstruction for F k k then each C is an
i
C i obstruction for F
i
Pro of First note that C for otherwise removing that comp onent from G is
i
a contradiction to the fact that G is an obstruction Thus we claim that each C is
i
an obstruction to a smaller family i e some F k where k k
CHAPTER FINDING FORBIDDEN MINORS
Assume that C is not an obstruction for F C the smallest family not
i i
containing C Not b eing an obstruction implies that there is a minor C of C such
i i
that C C But this implies that for the minor G C C of G
i i
G C C C C
i i
C C C C
G k
The existence of this minor contradicts the assumption that G is an obstruction for
F k So the connected graph C must also b e an obstruction
i
S
r
Corollary If G C is an obstruction for F k k then each C is an
i i
i
obstruction for F C i r
i
Pro of This follows by recursively applying the ab ove theorem
Many obstruction set characterizations are based on k parameterized lower ideals
It would b e surprising if the growth rate in the numb er of obstructions p er each k
is slow To conclude this subsection we presenttwo observations to substantiate this
claim
c
Observation If the numb er of connected obstructions f of F k is at least
k
one for all k then the total numb er of obstructions f of F k is greater than or
k
equal to the number of integer partitions of k
Pro of Because of prop erty ofF k weknowthatany obstruction O O F k
has O k By Corollary there exist disconnected obstructions O O
P
m
O where each O is connected and O k There is a one to one
m i i
i
corresp ondence with these non isomorphic obstructions and the number of integer
partitions of k
Example We can easily generate the number of integer partitions for various
n Hence the following table gives lower b ounds on the total numb er of obstructions
for F k
CHAPTER FINDING FORBIDDEN MINORS
n
counts
n
counts
To lead up to our next observation let F fG j G k g be any k
k
parameterized lower ideal that is de ned bysomeinteger function G p jGj
that is b ounded by some p olynomial function p Also let p G k denote the corre
sp onding graph problem that determines if G k where b oth G and k are part of
the input
Observation If p G k isNP complete then f jO F j must b e sup er
k k
P
p olynomial else the p olynomial time hierarchy PH collapses to
Pro of Yap see Yap has shown that the following are equivalentfor i
P P P P
a p oly or p oly
i i i i
P P
p oly p oly b
i i
P P
p oly p oly or PH p oly c PH p oly
i i
Nowiff is b ounded by a p olynomial then co p G k NP p oly byhaving
k
p olynomial advice consisting of the obstruction sets for all k p jGj That is
we can nondeterministically verify in p olynomial time for input G and k that some
obstruction O O F is a minor of G Sincep G k isNP complete we get
k
P P
co NP NP p oly p oly
An application of Yap s result given ab ove shows that
P P
p oly p oly PH p oly
P P
And this implies by another theorem of Yap that states that for i
P P P P
p oly p oly
i i i i
P P
for any i then the p olynomial So using the well known fact that if
i i
P P
time hierarchy collapses to level i we get PH unless there are a sup er
p olynomial numb er of obstructions for F k
CHAPTER FINDING FORBIDDEN MINORS
This observation can b e applied to most k parameterized lower ideals For ex
ample our k Vertex Cover and k Feedback Vertex Set graph families satisfy
the preconditions of the ab ove result Also since determining the genus of a graph is
NP complete the lower ideal k Genus fG j g enus G k g also b elongs in the
ab ove class
Searching via universal distinguishers
One approach for sp eeding up an obstruction set search is to try nd a smaller search
tree In fact we can do this by de ning a variation of our previously discussed
minimal t parse search tree
De nition An extension Z is a universal distinguisher for a t parse b oundaried
graph G if for every one step minor G of G wehave
G Z F and G Z F
A t parse G is universal minimal if there exists a universal distinguisher for it
Wehave found several initial lists of obstructions using this universal distinguisher
approach e g see Chapter The search tree restricted to universal minimal graphs
is smaller b ecause of the following observation
Observation With resp ect to a canonical congruence the set of universal
F
minimal t parses is a subset of the set of minimal t parses
The next lemma guarantees that this search approach is a viable one
Lemma The Pre x Lemma Lemma holds for universal distinguisher search
ing That is any pre x of a universal minimal t parse is universal minimal
Pro of First observe that every b oundaried obstruction has a canonic t parse rep
resentation If G Z is a b oundaried obstruction then G is universal distinguisher
minimal via Z
CHAPTER FINDING FORBIDDEN MINORS
The ab ove pre x prop erty means that our underlying obstruction set search soft
ware can b e used here with only a few alterations
It seems to b e harder in the computational sense to determine when a t parse
G is universal minimal or universal nonminimal The go o d news is that we can use
a slightly altered version of our random distinguisher searchtoshowthatat parse is
universal minimal Here for a t parse G we simply try to nd one extension Z such
that G Z is not in F while for eachminorG G G Z is in F As done for
m
the randomized algorithm given in Section we should minimize Z with resp ect
to G and F b efore checking if it is an universal distinguisher
The pro of of Lemma tells us that if we run enough random distinguisher
searches see Section then wewilleventually nd all the b oundaried obstruc
tions for F By using only this typ e of pro of technique there is a simple approxima
tion searchscheme that has the p otential of nding all the obstructions
Next wegive a concrete example of a universal minimal t parse
Example Let us reconsider the Vertex Co ver family of graphs that is
restricted to t parses for t The short t parse
n n n n
G
n
If H is anyoneof is universal minimal by the distinguisher Z
G s four non isomorphic one step minors wecan ndavertex cover of size two for
H Z see the squared vertices b elow As can b e seen the b oundaried graph G is
a pre x of the obstruction C
G Z extended minors
We further explore the idea of universal distinguisher searching in the next chap
ter under the more general framework of using second order congruences
CHAPTER FINDING FORBIDDEN MINORS
Using other nite ind ex congruences
One maywonder if we really need to use the canonical family congruence or can
F
we use another easy to program congruence for a lower ideal F The main result
of this section answers this question in the a rmative First let us formally de ne
what is meantby a nite state algorithm for anylower ideal F
De nition A nite state algorithm A congruence for anylower ideal of
F
graphs F i e a subset of the t parses satis es these four prop erties
t
The algorithm A is a mapping of t parses to a nite set of states which
F
corresp ond to the integers f rg for some r
If G and H are two t parse s that are not F congruent G H thenA G
F F
A H
F
If G and H are two t parses that are not memb ers of F then A G A H
F F
that is there exists an unique out of family state
If G and H are two t parses such that A G A H then for any z
F F t
A G z A H z
F F
The states of any nite state algorithm A represent a re nement of the canonical
F
family congruence Often these algorithms are written as dynamic programs
F
recall Section Also in the ab ove de nition the symbol denotes the implicit
congruence re nementof obtained from the states of A
F F
We can de ne t parse minimality in terms of For instance a t parse G is
minimal if for every prop er minor H G H G or A G and A H are
m F F
di erent states
Lemma If a t parse obstruction set search with substituted for termi
F
nates then the obstruction set for F of width t has b een found
Pro of Because of prop erty above any obstruction O of O F is minimal Also
since has only one out of family state prop erty all minimal out of family
CHAPTER FINDING FORBIDDEN MINORS
t parses are b oundaried obstructions The Pre x Lemma also holds for this de
nition of t parse minimality since a extensions are sup ergraphs and b the family
F is assumed to b e a lower ideal Therefore every pre x of O is minimal and will
not b e pruned
Notice that for the ab ove lemma termination of the search pro cess do es not rely
on the fact that the algorithm A has a nite numb er of states That is it do es not
F
use prop erty So it turns out that an in nite state algorithm or at least one not
proven to b e nite state satis es the premise and result of the lemma
For these non canonical congruences one handy result is missing This is the
ability to prove minimality via one step minors see Lemma It is p ossible in
fact to do a tree likesearch for any generic congruence and rede ne minimality
That is in this context a t parse G is said to b e minimal if and only if each one
step minor is distinguished from G However doing so maymake it easier for the
algorithm not to terminate or at the very least generate a larger than needed search
tree An exp erimental compromise is to try alternative minimality de nitions for
various minor steps up to some k
Finding uses of randomization
The metho ds of the previous two sections have something in common in that wemay
wanttoendasearchbeforeeverything is proven Let us call an approximation run
foralower ideal F any computation that yields a searchtreefor O F that still
has t parses that are not proven minimal or nonminimal That is wemay terminate
the program early where unknown t parse s remain We reuse pro ofs obtained on a
previous approximation run when we restart an obstruction set search for the same
lower ideal and width The topic of this section concerns how randomization can
b e used to improve the chances that rep eated runs progress towards nishing i e
wewant to reach a p oint where all t parses havebeenproven either minimal or
nonminimal Here wewant subsequent approximation runs to do di erentwork via
nondeterminism Doing this provides a b etter chance of nding new pro ofs
Dep ending on how hard it is to for the computer to obtain a pro of we store
CHAPTER FINDING FORBIDDEN MINORS
thestatusofa t parse e g minimal nonminimal noncanonic or irrelevant in a
centralized database to b e accessed in future searches For any t parse that has an
unknown status e g after a time out canonic attempt the searchmust assume
that the t parse is on a path to an obstruction e g in this case the t parse is assumed
to b e canonic
Wehave already seen randomization used in the random extension searches of
Section There are other subtle places that randomness may help One example
is for the algorithm that determines whether a t parse is canonic or not It is evident
that we do not want to do duplicate work during rep eated searches For example
rep eated CPU time should b e avoided if a t parse canonic run is inconclusive during
the rst time There are many op en problems regarding howwe can use randomization
to help make our obstruction search theory more practical
Chapter
The Implementation and
The Future
In this chapter we describ e our software for computing obstruction sets and discuss
some researchthatiscurrently b eing explored Our automated approachiscontinu
ally improving by this new theory
Using Distributed Programming
We b elieve that in some sense the most practical asp ect of our obstruction set search
theory is that it p ermits a distributive programming approach In the following
paragraphs we explain our e orts in exploiting this theme Figure at the end of
this section gives a visual p ersp ectiveofthemanytyp es of pro cesses that are involved
These key pro cesses are discussed b elow
Manager This pro cess is the main search tree engine of the system The manager
controls the database of status and pro of information regarding t parse s We
run the manager on a computer that contains a dedicated lo cal disk for the
database e g wehave access to a Sparc researchmachine at the University
of Victoria for this purp ose It is also preferable to have a large swap space
b ecause a skeleton of the search tree is kept in memory for the duration of
CHAPTER THE IMPLEMENTATION AND THE FUTURE
the search This tree includes record p ointers to the database for all t parses
enumerated The manager tells another pro cess called the dispatcher what
work needs to b e done and waits for results The manager grows the search
tree in a greedy fashion as work b ecomes available This strategy is dep endent
on the sequence order of pro ofs returned by the dispatcher These are pro ofs
of t parse minimality or nonminimality Also the manager is the only pro cess
that writes to the database Another pro cess called the lo cker issues read write
access tokens to pro cesses wanting access to the database
Dispatcher The dispatcher is the pro cess that distributes t parse work to the
worker clients It maintains a queue of work to b e done along with a queue of
idle worker pro cesses The work consisting of enumerated t parses with each
t parse initially having unknown status is senttoaworker using the standard
TCP IP so cket communication proto cols The worker pro cesses can b e any
where on the internet and not necessarily of the same computer architecture
e g we currently supp ort Sparcs Cray Y MPs and IBM s For commu
nicating pro cesses running on a machine with the same op erating system as the
dispatcher usually SunOS asynchronous I O is used otherwise synchronous
I O is used
Workers The workers consume most of the CPU resources during an obstruction
set search The main task for these pro cesses is to determine whether a t parse
is minimal or not using the metho ds of Section Some workers may also
check whether a t parse is canonic or not Since wehaveworkers compiled and
installed on multiple hardware platforms we use a sp ecial binary I O stream
ob ject a C class to facilitate communication b etween the dispatcher and
aworker The number of workers p ossible for an obstruction set searchis
practically unlimited ho wever our largest searches use only ab out machines
mostly Sparcs
Extractors These pro cesses are the user s interface to the distributive system
The extractors and browsers for interactive viewing are to ols that retrieve
information such as
CHAPTER THE IMPLEMENTATION AND THE FUTURE
The status of an individual t parse e g a partial set of one step minors
that have b een distinguished
A view of the current search tree
A dump of the b oundaried or ordinary minor order obstructions found
The output is pro duced in either text PostScript or binary form
The current statistics ab out howmany minimal and nonminimal no des p er
depth of the search tree This is used quite frequently in order to predict
how close a current obstruction set search is to completing
An archive or backup of the current search which is later used in a restart
For example this is used b efore a scheduled p ower outage
Wehave another typ e of extractor or more accurately called a controller that
allows us to p eek at the status of all system worker pro cesses This backdo or
into the dispatcher allows us to monitor the searchattheinternet level We
can manually kill o frozen connections through this backdo or
Software Summary
During our quest for a general purp ose system for computing obstruction sets we
have redevelop ed several key software comp onents In fact the current system was
restarted from scratchfouryears ago The rst version was dropp ed after one year
of e ort when we discovered that the automaton constructed by the metho ds given
in Section were to o big to b e practical
Some of our initial goals for a software pro ject is re ected in our system s acron ym
VLSI Automated Compilation System VACS The VACS research group at the Uni
versity of Victoria with Kevin Cattell and MikeFellows as the other two members
envisioned a system that could generate approximating automaton for real world en
gineering problems A metho d of using obstructions for doing this task is mentioned later in Section
CHAPTER THE IMPLEMENTATION AND THE FUTURE
DATABASE COMMANDS
#1
#n
Figure Schematic view of our distributed obstruction set software
CHAPTER THE IMPLEMENTATION AND THE FUTURE
Wenow illustrate the user s steps required to nd minor order obstructions using
our software We assume that the user is interested in a particular lower ideal F
Refer back to Section for more information on the various pro cesses mentioned
below
Create a problem le directory for the lower ideal F and the desired t parse
search width
Add source co de for the lower ideal F that consists of a family memb ership
algorithm and optionally any or all of the following nonminimal pretests a
nite index congruence or a testset There are general C mechanisms in
place for adding each of these comp onents If none of these pro of techniques
is provided the system will only use random distinguisher searches to prove
t parses minimal and may only compute a partial set of obstructions
Make a control le that contains key word descriptions of the problem This
should consists of at least the required run time parameters that are listed in
Figure
Start a database locker daemon on the machine containing the problem direc
tory This needs to b e started only once and resides for rep eated runs
Invoke the search script This csh script rst starts the manager pro cess
whichinturnspawns a dispatcher pro cess The dispatcher will create an inter
net so cket and store the currentportnumber chosen b y the op erating system
in an external le
Start as many worker pro cesses as needed If the machine do es not have access
to the problem directory then the internet p ort numb er has to b e manually
copied to a clone directory without the database on the remote machine
Monitor the progress of the searchwithvarious extractors and browsers
When the search completes run extractors to retrieve the obstructions for the
given search width and statistics of the current search The search is gracefully
terminated by the manager telling the dispatcher which in turn noti es all of
the workers to ab ort
CHAPTER THE IMPLEMENTATION AND THE FUTURE
const ProbInfo ParseTableEntry ProbInfo parseTable
problem name Required String
problem number Required Integer
directory Required Subpath
keyword Required String
pathwidth Required Integer
membership flag Optional Integer
use congruence Optional Boolean true
congruence flag Optional Integer
connected Optional Boolean true
tree search Optional Boolean false
conservative grow Optional Boolean true
extn edge weight Optional Integer
extn runs Optional Integer
extn tries Optional Integer
extn max length Optional Integer
extn skip Optional Integer
save nonmin proofs Optional Boolean true
congruence is tight Optional Boolean false
use testset Optional Boolean true
tilde minor search Optional Boolean false
max equal iso level Optional Integer
workers run canonic Optional Boolean false
load all nonminimal Optional Boolean false
limit to prefix Optional String
send PMP Optional Boolean true
univ dist search Optional Boolean false
pretest extensions Optional Boolean false
canonic timeout Optional Integer
Figure Some available run time options for obstruction set computations
CHAPTER THE IMPLEMENTATION AND THE FUTURE
Wenow brie y explain the optional run time parameters given in Figure for
our obstruction set searches The membership flag option is used to sp ecify a run
time option to the memb ership algorithm mainly used for testing purp oses or to pro
vide additional information ab out the currentlower ideal The use congruence
option tells the workers that wehave implemented a re nement of the canonical
congruence and to use it The congruence flag option is available for di erent
workers to run the dynamic programming congruence in di erent mo des Using this
option and the memb ership ag the user can quickly change asp ects of the search
without having to recompile the program The connected option noti es the man
ager to prune the search tree at the disconnected t parses see Section The
tree search option allows one to restrict the domain of the search to trees i e
treewidth one graphs of b ounded pathwidth Using conservative grow as an op
tion tells the manager to nish the search tree at level h all t parse strings of length
h t b efore enumerating t parse s at level h The ve options extn edge
weight extn runs extn tries extn max length and extn skip allow
the user to control the random distinguisher searches Here one sp eci es the edge
densit y as a p ercentage limits the attempts for proving that a t parse G is mini
mal b ounds howmany times the search tries to distinguish a sp eci c minor G from
G sp eci es the maximum length numb er of op erators for each extension and states
how many op erators are app ended to Z b efore G Z is tested for memb ership in F
The save nonmin proofs option allows the user to save in the family database all
nonminimal t parse pro ofs These pro ofs are not needed when the search uses only a
random distinguisher searches as the only t parse pro of technique see Section
The congruence is tight option noti es the workers that the implemented con
gruence is in fact the canonical congruence for F The use testset option tells
the workers that a testset for is available and can b e either loaded from disk or
F
enumerated at run time The tilde minor search option turns o the one step
minor mo de for proving t parses minimal The max equal iso level option is
used only during a restart of the search Here the pro ofs from a backup database
are loaded unconditionally for all t parses shorter in length than this iso level any
t parse not in the database up to this level is assumed to b e non canonic The
workers run canonic option allows the manager to request to the dispatcher
CHAPTER THE IMPLEMENTATION AND THE FUTURE
that the workers should help prove whether a t parse is canonic or not recall the
two canonic stages mentioned in Section The load all nonminimal op
tion is used to cancel the default mo de of retrying random distinguisher searches
for assumed nonminimal t parses this is used in a restart of an approximation
run The limit to prefix option allows the searchtostartatany subtree of
the searchtree we used this for our Feedback Vertex Set computations The
send PMP option tells the workers to send backall Partial Minor Pro ofs for each
t parse G that they handle e g a pro of is partial if all of the one step minors of
G have not b een distinguished from G The univ dist search option turns on
the universal distinguisher search metho d that was presented in Section The
pretest extensions option is for the random distinguisher searches Here each
extension Z must fail every direct nonminimal pretest that is installed for the current
lower ideal F Finally the canonic timeout option allows the impatient user to
sp ecify the maximum amount of CPU time that can b e sp ent on trying to prove that
a t parse is non canonic If the timeout is reached then the system assumes that the
t parse is canonic for the integrity of the search
F or our obstruction set searches we ran our manager and dispatcher pro cesses
on a Sparc workstation This machine has a relatively small amount M of
dedicated lo cal disk space This drivewas mainly used to store the search database
However this machine has M RAM of internal memory which allows for less disk
swapping during our larger obstruction set searches We concurrently ran workers at
b oth the University of Victoria only Sparcs and Los Alamos National Lab oratory
Sparcs Cray Y MPs and IBM s
The following Table lists our main software directories with the line counts
for the source co de Not included is a few thousand lines of miscellaneous co de e g
sp ecial source les to interface the LEDA graph library The size of the executable
program is K after b eing stripp ed of debugger co de this executable w as pro duced
by a Sparc AT T CC compiler We also haveover one thousand lines of shell
script co de where the average length of each script is ab out thirty lines These
scripts were written in Perl csh or ksh In fact an ASCI I version of Table was
generated by a simple csh script that calls the UNIX sed and awk utilities
CHAPTER THE IMPLEMENTATION AND THE FUTURE
Table Source co de breakdown by area for our VACS software sub directories
Header Files Source Files
c lines h lines Directory c lines Directory
inc src automata
inc array src browser
inc automata src tparse
inc tparse src tparse algorithm
inc tparse i o src tparse i o
inc tparse algorithm src family
inc browser src family testset
inc family src lesystem
inc family testset src general
inc general src graph
inc graph src graph algorithm
inc graph algorithm src graph generate
inc graph i o src graph i o
inc isomorphism src graphplay
inc search src isomorphism
inc set src i o
inc vacs src lo cker
src pro cess
src search
C template co de is included here
src searchmain
src searchno de
src sp ecial c lines h lines and c h lines
CHAPTER THE IMPLEMENTATION AND THE FUTURE
The main engine for the VACS pro ject is built and in op eration The computa
tional results of this dissertation are our initial achievements regarding our systematic
metho d for characterizing lower ideals However likeinany ma jor software pro ject
there are several rough sp ots that need to b e smo othed out The next section men
tions a few areas of researchthatwe are currently pursuing
Future Research Goals
Wenow b egin with some obvious research topics that wewant to explore rst These
areas of research all deal with making obstruction set computations more practical
i e continuing the theme of Section
Implement the b ounded treewidth search One of our rst tasks is to com
plete the pro of for our Feedback Vertex Set obstruction set The easiest
mechanical way of doing this is to patch the current system with a treewidth
t parse search although Chapter gives strong arguments that there maybean
alternate pro of We note that searching through graphs of b ounded treewidth is
not straightforward For example our enumeration scheme presented in Chap
ter is not as e cient for the treewidth case as it was for the pathwidth case
Thus wemay need results from the following research area to quickly purge
t parse redundancies
Develop b etter canonic algorithms for graphs of b ounded combinatorial width
Currently weha ve a practical canonicity algorithm for small t parse s For
some of our larger lower ideals we noticed that some t parses with as few
as vertices caused the algorithm to stall This is why a canonic time out
mechanism was installed Since graph isomorphism is p olynomial time when
restricted to graphs of b ounded treewidth pathwidth we also susp ect there
is a guaranteed p olynomial time algorithm to test if a t parse is canonic We
can not rule out the p ossibility that another op erator set is more suitable for
uniquely representing the set of partial t trees t paths
CHAPTER THE IMPLEMENTATION AND THE FUTURE
Use a virtual b oundary size with resp ect to the b oundary minor order We
have noticed that our current search trees haveseveral long branches that are
the result of the b oundary minor order b eing weaker than the minor order
For example we nd b oundary minor obstructions with isolated b oundary ver
tices even though we are pruning the search at disconnected t parses See
Section To reduce the size of such a search tree one obvious idea is to
rede ne the b oundary minor order Supp ose we consider graphs with a variable
sized b oundary up to some maximum size Here the deletion of an isolated
b oundary vertex could b e taken as a minor Also edge contractions b etween
b oundary vertices would b e legal minors Since there are more minors for a
given graph G in this new framework the chances of nding a minor congruent
to G with resp ect to improves Thus there will b e fewer minimal t parses
F
in the search tree Using virtual b oundaries is promising except for the fact
that our pre x prop erty see Lemma may no longer b e valid
Reuse minimal t parses for higher searchwidths If one has an appropriate
canonic scheme one can reuse minimal b oundaried graphs at width t for a
width t search p erhaps by simply inserting a new vertex op erator at the
b eginning of each minimal t parse This idea follo ws from the observations
minimality is satis ed if and only if all minors are distinguished by some ex
tension and the set of graphs with a larger width is a bigger p o ol sup erset
of p otential distinguishers It should b e easy to implemented this computation
saving feature
Do recursive minor checks For any nite index congruence it is p ossible
to showthatat parse G is nonminimal by nding a congruent b oundaried
minor that is more than a one step minor of G Herewe construct a p oset of
b oundaried minors similar to Figure with G at the top and the smallest
n n n
t parse t at the b ottom Wecheck the equivalence classes of all
prop er minors against G s equivalence class b eginning at the top and pro ceeding
down in a breadth rst fashion Using recursive minor checks for is
F
sup er uous by Lemma
CHAPTER THE IMPLEMENTATION AND THE FUTURE
Adapt our searchtechnique to handle other well partial orders So far wehave
concentrated on the minor order There are other interesting graph orders
such as the immersion order see Section that our search pro cess may
b e adaptable However there are some details that need to b e resolved For
example wehave to handle edge lifts across the b oundary in the b oundary
immersion order with resp ect to a lower ideal s canonical congruence
Two more prop osed and on going research areas which are based on additional
theoretical ideas are presented in the following subsections
Second order congruence research
Another technique to reduce the search space for obstruction set computations is to
use a second order congruence in replace of the canonical family congruence
F
Recall that the quanti ers and are read there exists and for all resp ectively
De nition The canonical second order congruence for a lower ideal F de
noted by is de ned on nite sets of t b oundaried graphs by if S and S are
F
t b oundaried graphs then S S if and only if for every t b oundaried graph Z
F
X S X Z F X S X Z F
A non canonical second order congruence for F is an equivalence relation de ned
on nite subsets of t b oundaried graphs for which S S implies S S
F
Wenowpoint out that second order congruences are related to our universal
distinguisher search metho d that was presented in Section Let G G denote
m
aproper minor of a t b oundaried graph G i e G G and G G For any
m
Gg and S fG j G Gg t b oundaried graph G consider these twosetsS f
m
We claim that G is nonminimal with resp ect to a universal distinguisher search if
and only if S S That is for every t b oundaried graph Z
F
G Z F G G G Z F m
CHAPTER THE IMPLEMENTATION AND THE FUTURE
Or in a more familiar form there do es not exist a t b oundaried graph Z such that
G Z F and G G G Z F
m
And b ecause F is a lower ideal we easily note in the other direction for any t
b oundaried graph Z
G G G Z F G Z F
m
The ab ove logic leads us to the next observation
Observation If wehave a second order congruence for a minor order lower
ideal F then we can easily run a universal distinguisher search
One of our motivations for using second order congruences is the p ossibilityof
detecting when to stop the search without a prior knowledge of the largest combina
torial width of an obstruction In fact the following main result of CDDF uses
stopping conditions along with a factoring result for b ounded pathwidth similar to
the one given in Theorem of Chapter
Theorem Supp ose the following are known for a minor order lower ideal F
A decision algorithm for F
A decision algorithm for a terminating second order congruence for F
Then the obstruction set O F can b e computed by an algorithm that uses the ab ove
two algorithms as subroutines
We p oint out that if a second order congruence is not known to satisfy our ter
minating de nition we still may b e able to detect when to stop the search Further
more wewould like to answer this question How can we guarantee that a signal to
stop will b e detected so on after the last obstruction has b een computed Wewould
like to stop the search after the minimal graphs of width t have b een searched
whenever the largest obstruction has width t
CHAPTER THE IMPLEMENTATION AND THE FUTURE
Approximation algorithms based on partial obstruction
sets
This dissertation has primarily b een interested in computing complete characteriza
tions of lower ideals by forbidden minors We realize however that there are some
potential uses for partial lists of obstructions We think that there is a future in
building graph theoretic approximation algorithms from a selected subset of the ob
structions Wehave already built a go o d software foundation for such an automated
algorithm compiler
It is susp ected that for any minor order lower ideal F most of the non family
graphs are ab ove a small numb er of the obstructions O F It is well known that the
family of planar graphs is almost the same as the set of graphs that excludes just the
one Kuratowski obstruction K For example we can prove the following known
result
Observation Any three connected non planar graph with at least six vertices
contains K as a minor
Another promising example was recently exhibited in the VLSI design area by
the work of Langston and Ramachandramurthi LR Ram Their simple metho d
of using just one or a few of the track gate matrix layout obstructions or equiv
alently the Pathwidth obstructions for a memb ership algorithm has out p er
formed the other b est known heuristics for the problem Note that this researc his
based on a small p ercentage of the complete set of the obstructions recently
proven complete by Kinnersley KL
These two observations suggest that go o d approximating automata may b e con
structed by using partial obstruction sets
One maywonder what obstructions are go o d candidates for approximating any
sp eci c lower ideal Actually thistechnique maywork for almost lower ideals
to o We know of three classes of obstructions that may b e useful
The obstructions with the smallest number of vertices or edges
CHAPTER THE IMPLEMENTATION AND THE FUTURE
All of the obstructions that are b ounded by a small combinatorial width
The obstructions that o ccur frequently via our random distinguisher searches
It would b e interesting to do some exp erimental computing to see how close in prac
tice a family of graphs de ned by excluding one of these classes is to a targeted lower
ideal The b ene t of using the second class of obstructions i e those of b ounded
combinatorial width is that wedonothave to compute the complete set of obstruc
tions Also related to the third class it is natural to assume that if a distinguisher Z
for a t parse G is easily found then the obstructions b elow G Z are go o d candidates
It is exp ected that the ab ove three classes of obstructions overlap
The easiest way to use a partial obstruction set for some family F is to write quick
minor order tests fM g for each obstruction O i r The approximating
i i
algorithm for an input graph G would return no if any obstruction O is found as a
i
minor using M Otherwise it would return yes that G isamember of F Note
i
that this algorithm has only one sided errors This happ ens whenever it returns yes
when the real answer is no
Unfortunately the ab ove approachisnotvery automated This is b ecause there
is no general e cient algorithm that tests if any xed graph O is a minor of another
i
Recall that the Rob ertson Seymour O n algorithm has large hidden constants
However when the input is restricted to graphs of b ounded combinatorial width there
is hop e In this case we can build a minor testing automaton for a xed graph O
i
using the metho ds of Section A testset T for O consists of b oundaried graphs
i i
obtained from pieces of sub divided O Here the constructed automaton M accepts
i i
t parses that contain O as a minor
i
Another metho d for generating approximating automata is to use pieces of all the
r
obstructions as testsets Using this approachwe build one automaton for all of O
i
i
for a lower ideal F instead of one automaton for each obstruction Since obstructions
r
usually share the same substructures this testset T T should b e of manageable
i
i
size However it may b e easier to build a nondeterministic automaton M that accepts
r
graphs in the union of the minor containment families O or the intersection
i
i
r
of the single obstruction lower ideals O from the deterministic automata
i
i
r See Figure One mayeven convert M to a deterministic M i i
CHAPTER THE IMPLEMENTATION AND THE FUTURE
automaton and then minimize
For the b ounded width domain wemay elect to use the set of easily proven
minimal t parses as an approximating testset since these in most cases are already
pieces of obstructions
q
q
f
DFA M for O
q
f
q
accept state
start state
q
q
f
DFA M for O
r
r
q
q
f
DFA M for O
r r
Figure Building an NFA that accepts the union of minor containment families
Part II
Obstruction Set Characterizations
Chapter
Vertex Cover
Wemaketwocontributions in this chapter First we present a linear time algorithm
that determines the size of the minimum vertex cover for graphs of b ounded path
width partial t paths This algorithm has the imp ortant prop erty of b eing minimal
de ned in Chapter and reviewed later in Section It is this prop ertywhich
allows us to compute the second contribution of this chapter the obstruction sets for
the rst ve parameterized family instances of the vertex cover problem
Intro duction
The general problem of determining if a graph has a vertex cover of size k with k
part of the input is well known to b e NP complete GJ However several NP
complete problems have p olynomial time parameterized versions In any of these
instances a problem sp eci c integer k is held constant The vertex cover problem is
one such example where we are interested in determining if an input graph has a ver
tex cover of size at least k where k is not part of the input The brute force approach
k
of checking all k subsets of the vertices gives a crude O n algorithm Alterna
tively if a tree or path decomp osition of b ounded width is available determining the
minimum vertex cover of a graph can b e done linear time ALS In addition to our
obstruction set characterizations we present a practical nite state algorithm for the
setofgraphswithavailable path decomp ositions of b ounded width Furthermore by
CHAPTER VERTEX COVER VC
the observations path decomp ositions for xed k can b e found in linear time see
Bo d c Klo a and a pathwidth b ound exists for the family see Theorem
wehave a linear time algorithm Interestingly a direct parameterized algorithm is
presented in DF with the same complexity
The rest of this chapter is organized as follows The remaining part of this section
formally de nes the vertex cover problem and intro duces results and notation used in
this chapter Section presents our general vertex cover algorithm and its minimal
nite state variation for b ounded pathwidth graphs Section contains results that
reduce the amountofwork needed to compute the obstructions sets Section
presents the obtained obstructions sets and we conclude in Section with some
comments regarding the related k IndependentSet and k Clique families
The general vertex cover decision problem is de ned as follows see GJ
Problem Vertex Cover VC
Input A graph G V E and a p ositiveinteger k jV j
Question Is there a subset V V with jV j k suchthat V contains at least one
vertex from every edge in E
AsetV in the ab ove problem is called a vertex cover for the graph G The
family of graphs that havea vertex cover of size at most k will b e denoted by
k Vertex Cover For a given graph G letVC G denote the least k suchthatG
hasavertex cover of cardinality k
Example A graph in the Vertex Cover family is displayed b elow Notice
that no edges remain when the set of blackvertices of the example or indeed any
vertex cover is removed from the graph
CHAPTER VERTEX COVER VC
Finding the smallest vertex cover of a graph is imp ortant for many applications
For a simple example consider a communications system that is mo deled as a graph
where vertices are computers and edges are the transmission lines A minimum vertex
cover of the graph indicates the smallest numb er of computers that need sp ecial
monitoring software for all of the connections of the system
The next result stating that these parameterized vertex cover families can b e
characterized by forbidden minors is well known
Lemma The graph family k Vertex Cover is a lower ideal in the minor
order
Pro of Assume a graph G V E k Vertex Cover has a minimal vertex cover
V V IfH Gnf u v g for some u v E edge deletion then V is also a vertex
cover for H Likewise if u V is an isolated vertex of G V also covers H G nfug
vertex deletion For anyedge u v E observe that jfu v g V j Let w be
the new vertex created from u and v in H G u v edge contraction Clearly
V V fw g nfu v g is a vertex cover of H with cardinality at most k Since any
minor of G can b e created by rep eating the ab ove op erations k Vertex Cover is
alower ideal
A Finite State Algorithm
In this section wegive a practical nite state algorithm for the vertex cover prob
lem on graphs of b ounded pathwidth in t parse form This linear time algorithm
is a dynamic program that makes a single left to right scan of the input t parse
G g g g The computational pro cess resembles a nite state automaton
n n
in that it accepts words over the op erator alphab et Let m b e the currentscan
t
p osition of the algorithm on input G Thestate table at op erator g is indexed by
n m
t
each subset S of the b oundary These di erententries are de ned as follows
V S minfjV j V is a vertex cover of G and V S g
m m
CHAPTER VERTEX COVER VC
Two imp ortant observations ab out the state table are
For each b oundary subset S V S is a non decreasing sequence of non
m
negativeintegers as m increases
For any b oundary subset S and any b oundary vertex i S either V S
m
V S fig orV S V S fig
m m m
The algorithm given in Figure starts by setting the sizes for the minimal
n n n
vertex covers on the edgeless graph G t for all subsets S of the
t
initial b oundary
The typ e of the op erator g a vertex op erator or an edge op erator determines
m
ho w the state table is up dated during the scan The up date of an entry for a sp eci c
subset of the b oundary S is further broken up according to the relationship b etween
S and the op erator These transitions are describ ed in cases of Figure
When the algorithm reaches the end of the t parse it has computed the mini
mum number of vertices needed for a vertex cover of G This is b ecause the entry
n
V contains the size of the smallest vertex cover that contains the subset of the
n
boundary As this is an empty condition V is the size of the smallest vertex cover
n
in G
n
Theorem For any t parse G g g g the algorithm in Figure cor
n n
rectly computes VC G
n
Pro of If G is the empty graph then only steps I and I I I are executed and the correct
result of VC G is returned Assume that the algorithm is correct for all pre x
graphs of length m and less Weshow that cases of step I I correctly up date the
state table V S forS
m
n
Case g i and i S
m
Let V b e a witness vertex cover for V S Since the new vertex created by g
m m
do es not add any edges V is a vertex cover for G Since V S for G and
m m
i S V S for G Therefore V S V S
m m m
CHAPTER VERTEX COVER VC
I For m t set for every S
V S jS j
t
II For t m n do the following cases
n
Case vertex op erator i and i S
V S V S
m m
n
Case vertex op erator i and i S
V S V S nfig
m m
Case edge op erator ij where i S or j S
V S V S
m m
ij where i S and j S Case edge op erator
V S minfV S fig V S fj g g
m m m
III The size of the minimum vertex cover of G is
V
n
Figure A general vertex cover algorithm for t parses
CHAPTER VERTEX COVER VC
Let V beawitnessvertex cover for V S Since V is minimal the isolated
m
vertex created by g is not in V Thus V is a vertex cover for G Since the
m m
prop erty V S is preserved V S V S
m m
n
Case g i and i S
m
Let S S nfig and V b e a witness vertex cover for V S Now W V fig
m
is a vertex cover for G such that W S So V S V S
m m m
For the other direction let V b e a witness vertex cover for V S Since the
m
new b oundary vertex i do es not help in anyvertex cover of G V V nfig is a
m
vertex cover for G such that V S Hence V S V S
m m m
Case g ij where i S or j S
m
Let V beawitnessvertex cover for V S Since the b oundary is not changed
m
by the edge op erator g and i S or j S V also covers the edges of G
m m
Thus V S V S If V is a vertex cover of G with i S or j S then
m m m
V also covers the edges of G SoV S V S
m m m
Case g ij where i S and j S
m
Let S S fig and V b e a witness vertex cover for V S Since V S
m
is a vertex cover for G asvertex i is in V V S V S Likewise if
m m m
S S fj g then V S V S So V S minfV S fig V S fj g g
m m m m m
Let V b e a witness vertex cover for V S Since i j is an edge either i V
m
or j V Thus V S fig or V S fj g If V S fig then V is a
vertex cover for G and so V S fig V S Otherwise V S fj g and
m m m
g V S Therefore minfV S fig V S fj g g V S V S fj
m m m m m
The following lemma shows that we can limit the vertex cover memb ership algo
rithm to a nite numb er of p ossible con gurations when testing for memb ership in
k Vertex Cover
Lemma For any xed integer k as an upp er b ound the algorithm given in
Figure can b e converted to a nite state algorithm A
k
Pro of Weshow that for xed k there are only a nite numb er of p ossible states
Consider the state table entry for a b oundary subset S IfV S b ecomes k for i
CHAPTER VERTEX COVER VC
some i then the monotonicityof V S guarantees that V S k for all j i
m j
As we are only interested in knowing whether or not there exists a vertex cover of
size k containing S we can restrict V S tobeinf k k g As there are
m
t
t
entries in the state table the numb er of states is b ounded by k
Tomake the parameterized algorithm A for k Vertex Cover change any
k
up date function V S f V in the four cases with V S min f V k
m m m m
It is straightforward to verify that this mo di ed algorithm correctly computes the
same state table except that anyentry greater than k is replaced by k
We de ne the nal state of a t parse G denoted by V to b e the state table
G
state vector V V S when the nite state algorithm A terminates For
m m k
each b oundary subset S let V S denote the S entry of V IfG and H are t parses
G G
and V V it follows immediately that for any op erator string Z wehave
G H
t
Z F H Z F for all Z That V V This in turn implies that G
G Z H Z
is G and H agree on all extensions When the converse of this prop erty holds the
nite state algorithm is minimal Formally minimality is satis ed if
for all Z G Z F H Z F V V
G H
t
Toshow that our nite state algorithm is minimal weneedtoshow that if G
and H are t parse s and G and H agree on all extensions then V V We will
G H
show the contrap ositive that is if V V then there exists an extension Z such
G H
that G and H do not agree on Z that is there exists Z such that either G Z F
and H Z F or G Z F and H Z F
Before proving that this k Vertex Cover algorithm is minimal weneedthe
following lemmathatprovides us with a b oundary vertex for building sucha t parse
extension Z
Lemma If G and H are t parses suchthatV V then there exists
G H
an S i e a prop er subset of the b oundary suchthat V S V S
G H
Pro of Assume that V V is the only di erence in the state table The
G H
following three facts
CHAPTER VERTEX COVER VC
V nfig V nfig for all i
G H
V nfig V V nfig for all i and
G G G
V nfig V V nfig for all i
H H H
imply that
V V nfig V nfig V forall i
G G H H
and
V V nfig V nfig V for all i
G G H H
After combining the ab ove V V V So without loss of
H G H
generality assume V V d From this identity and facts and
G H
ab ove also see the partial state tables b elow wemust have V V nfig d
G G
for all i
graph G graph H
V d V d
G H
d d
V nfig or V nfig or
G H
d d
This can happ en if and only if each of the b oundary vertices of G are attached to
some non b oundary vertex If not then a vertex cover V of G would havea
redundantvertex i The vertex cover created by eliminating vertex i from V
contradicts the value of V nfig However sucha graph G do es not exist since the
G
last vertex op erator can only haveboundaryvertex neighbors Thus we can conclude
that V V or there exists a S such that V S V S
G H G H
Theorem For the k Vertex Cover family the nite state algorithm A is
k
minimal
Pro of Let G and H be t parses As discussed ab ove we show that if V V
G H
then there exists an extension Z such that G and H do not agree on Z Note that
the theorem holds trivially if either one of G or H is not in F k Vertex Cover
CHAPTER VERTEX COVER VC
by the empty extension Z If b oth G F and H F then V V
G H
k k k since V k implies V S k for all S
G G
So supp ose that V V Then without loss of generality there is a b oundary
G H
subset S with minimum cardinality such that V S V S k Lemma
G H
guarantees that S
Let fv v v g b e the b oundary vertices in S Pick a b oundary vertex i S
jS j
and any other b oundary vertex j i Construct an extension Z as follows
k V S times
H
z
n n n n n n n
Z i iv i iv i iv i j ij i j ij
jS j
The extension Z essentially forces the b oundary vertices S to b e covered while
adding k V S isolated edges Now VC H Z is given by V S k
H H
V S which equals k andso H Z F However VC G Z isV S
H G
k V S V S k V S k andso G Z F Therefore
H H H
the t parses G and H do not agree on all extensions
to the t parse Example Table shows the application of the algorithm A
k
given earlier in Example on page As can b een seen by examining the graph in
Example a minimum vertex cover has cardinality which equals V
The VC Obstruction Set Computation
For k Vertex Cover two ingredients su ce to compute their obstruction sets
First we need to know a b ound on the pathwidth or treewidth of the obstruc
tion set In general sucha boundalways exists since the obstruction set is nite
Given such a b ound we can compute all of the obstructions by restricting our search
to a xed pathwidth Second wewant a minimal nite state algorithm that op erates
on t parses An overview of how the algorithm A is used is given in Section
k
For vertex cover wehave b oth of these ingredients A minimal nite state algo
rithm was describ ed in the previous section and a pathwidth b ound is shown later
in this section The nite state algorithm is also used as a memb ership algorithm
CHAPTER VERTEX COVER VC
Table Vertex cover state tables computed columns for Example
m
n n n
S g
m
f g
f g
f g
f g
f g
f g
f g
In short the input output combination used to compute the obstruction set for
a particular k Vertex Cover lower ideal is as follows
input pathwidth t
minimal nite state algorithm for k Vertex Cover
output obstructions of pathwidth t
The following sequence of results showthevalue of t that is needed to get the
complete set of obstructions O k Vertex Cover fork Vertex Cover and why
we can restrict our search to connected graphs
We rst need an upp er b ound on the pathwidth of the obstruction set That is
we need a result of the form if G O k Vertex Cover then G has pathwidth at
most k For vertex cover such a b ound is easily obtained We rst showthatthe
family k Vertex Cover is contained in the family k Pathwidth Thisresultis
ver iscontained in well known vL It follows from this that O k Vertex Co
k Pathwidth
Theorem The pathwidthofanymember of k Vertex Cover is at most k
CHAPTER VERTEX COVER VC
Pro of For a given graph G of k Vertex Cover letV b e a subset of of the vertices
of size k that covers all edges Denote the order of G by n Let the vertices V of
G b e indexed by n with the vertices V n V coming rst We claim that
fX j i n k g where X V fig is a path decomp osition of G
i i
S
X V Let Since every vertex is either in V or is in V n V wehave
i
i n k
u v beanedgeof G Since V is a vertex cover without loss of generality assume
u V If also v V then any subset X contains b oth u and v Otherwise v must
i
b e indexed b etween and n k and the subset X contains b oth u and v Finally
v
note that for any i j n k wehave X X V i e the interp olation
i j
prop erty is satis ed Thus wehave a path decomp osition of pathwidth k
The ab ove theorem cannot b e improved as the complete graph K with path
k
width k is a member of k Vertex Cover
Corollary If G O k Vertex Cover then the pathwidth of G is at most
k
Pro of For anyedge e u v E G let G G n e SinceG is an obstruction
for k Vertex Cover G k Vertex Cover and hence VC G k by Theorem
Let V b e a witness vertex cover for G Now V V fug is a vertex cover for
G of order at most k Therefore by Theorem again the pathwidth of G is at
most k
The numb er of obstructions we need to nd can b e reduced by some straightfor
ward observations The follo wing observations are sp ecial cases of the more general
results found in Section regarding disconnected pruning
Observation If C and C are anytwo graphs then VC C C VC C
VC C
Observation If O C C is an obstruction for k Vertex Cover then C
and C are obstructions for k Vertex Cover and k Vertex Cover resp ectively
for some k k k with k k k
CHAPTER VERTEX COVER VC
Hence we can restrict our attention to connected obstructions an y disconnected
obstruction O of k Vertex Cover has VC O k and is an union of graphs
S
k
from O i Vertex Cover
i
Example Since K is an obstruction for Vertex Cover and K is an
obstruction for Vertex Cover the disconnected graph K K is an obstruction
for Vertex Cover
The VC Obstructions
In essence our theory allows us to compute for any t all of the obstructions that
have pathwidth at most t However tractability problems arise as t increases As
shown in the preceding section we need to use t parses t k to obtain all of
the obstructions for k Vertex Cover
A brief description of the k Vertex Cover obstruction set computation is now
stated The set of all t parse s can b e viewed as a tree recall Section in which
the parentofa t parse G of length n is the length pre x of G of length n The
n n n
ro ot of the tree is the empty graph t The minimality of the nite
state algorithm allows us to directly compute a pruning rule for the tree That is a
t parse G is minimal if and only if it is not congruenttoany of its one step minors
Any obstruction minimal leaf of this search tree that has all of its minors
m
order in k Vertex Cover is a memb er of the obstruction set That is we just
check that b oundary edge contracted minors fall into the family
A summary of our obstruction set computations for various k Vertex Cover
families is shown in Table The total graphs column shows the size of the pruned
tree describ ed ab ove In the minimal graphs column the number of internal graphs
plus obstructions is shown that is the lea ves that are not obstructions have not b een
counted The growth rate of the tree can b e seen to b e extremely high as k and
h tree for the Vertex Cover hence the pathwidth t increases The revalent searc
runisshown in Figure for pathwidth
Besides the single obstruction K for the trivial family Vertex Cover the
CHAPTER VERTEX COVER VC
Table Summary of obstruction set computation for vertex cover
Minimal Total Connected Total
Elapsed
k
time graphs graphs obstructions obstructions
seconds
seconds
minutes
hours
days
connected obstructions for k Vertex Cover k are shown in Figures
Some patterns b ecome apparent in these sets of obstructions One such easily
proven observation is as follows
Observation For the family k Vertex Cover b oth the complete graph
K and the cycle C are obstructions
k k
Indep endent Set and Clique Families
Now consider the following related families of graphs where n denotes the order of a
graph and k is any non negativeinteger
k Clique f graphs with maximum clique k g
k IndSet f graphs with maximum indep endentset k g
k Clique f graphs with n maximum clique k g
k IndSet f graphs with n maximum indep enden tset k g
In fact one of these families i e k Clique is the same as our studied
parameterized vertex cover families GJ
Lemma The two graph families k IndSet and k Vertex Cover are
identical
CHAPTER VERTEX COVER VC #20,23 #100,23 #39,13 #38,12 #250,23 #169,13 #78,12 #244,3 #280,23 #274,3 #300,23 #239,13 #37,03 #296,02 #158,12 #272,1 #71,0 Nonminimal out-of-family #295,01 #293,2 #232,1 #155,01 width #226,02 #142,1 #68,12 #225,01 #138,12 #67,03 #130,23 #129,13 Minimal out-of-family (Obstruction) #128,12 #16,02 with graphs of path #210,23 #209,13 #5,01 #0,[0,1,2,3] ver #208,12 #270,23 #127,03 #268,12 tex Co #201,0 #266,02 #31,0 #265,01 Ver #126,02 #65,01 Nonminimal family member #199,13 #196,02 h tree for #260,23 #259,13 #258,12 #122,1 #257,03 #195,01 #256,02 #288,12 #287,03 #251,0 Minimal family member #285,01 #120,23 Figure Actual searc #190,23 #57,03 #111,0 #26,02 #11,0 #186,02 #107,03 #51,0
CHAPTER VERTEX COVER VC
Pro of It is easy to see that if I is a maximal indep endent set of a graph G V E
then V n I is a vertex cover Also if V is a minimal vertex cover then V n V is an
indep endent set
Part of the next result has b een proven earlier in Lemma This one illustrates
another metho d for proving graph families are lower ideals
Lemma The family k Vertex Cover k IndSet isalower ideal in the
minor order for any k while the families k Clique k IndSet and k Clique
are not lower ideals for all k k and k resp ectively
Pro of Let G V E b e a graph in k IndSet with maximum indep endent
set I Ifv is an isolated vertex of G then it is also in the set I Thus deleting v
from G preserves the invariant n maximum indep endent set Deleting anyedgee
can only increase the maximum indep endent set while n stays xed so G nfeg
k IndSet Let G b e the result of contracting an edge e u v of G and let I
b e a maximum indep endent set of G Also let w denote the new vertex of G created
by the edge contraction If jI j jI j then we get a contradiction to the fact that I
g fug is was maximum To see this consider these two cases If w I then I nfw
a larger maximum indep endent set for G Ifw I then I itself is a larger maximum
indep endent set for G SinceI is an indep endent set at most one vertex u or v can
be in I So going from G to G I I nfu v g is an indep endent set for G Thus
jI j jI j jI j Finally since the value of n decreases by exactly one during an
edge contraction G k IndSet Wehaveshown that all minor op erations are
closed with resp ect to k IndSet
Now consider the family Clique and a member C the cycle of length four
Contracting anyedgeofC pro duces C K whichisnota member of Clique
Similar examples suchas K with a sub divided edge show that k Clique is not
k
alower ideal for all k The only graphs in the family Clique are the isolated
graphs unions of K s and the only minors by isolated vertex deletions havea
maximum clique b ounded ab oveby So Clique isalower ideal
As p ointed out in the rst paragraph the maximum indep endent set can increase
with edge deletions It follows that k IndSet is not a lower ideal for all but the trivial
family IndSet i e the empty graph has no deletable edges
CHAPTER VERTEX COVER VC
For the family k Clique k consider a graph member G k K K
k
Note that G has n k vertices and a maximum indep endent set size of k Deleting
an edge from G causes an increase in the invariant n maximum clique For k
consider the graph G K K The graph created by deleting the edge from G has
n and maximum clique of so Clique is not a lower ideal in the minor
order Likewise since K is a member of Clique butany minor created bya
single edge deletion is not the family Clique isnotalower ideal
In view of the previous results we can characterize the k IndSet families
in terms of the nite k Vertex Cover obstruction sets The other remaining few
minor order lower ideals are trivial to characterize The reader should b e able to verify
that O Clique fK g O Clique fK g and O IndSet fK g
K C K C
Figure Connected obstructions for and VertexCover
K C
Figure Connected obstructions for Vertex Cover
C
K
Figure Connected obstructions for Vertex Cover
C K
Figure Connected obstructions for Vertex Cover
Chapter
Feedback Vertex Edge Sets
In this chapter wecharacterize twotyp es of simple graph families The rst family
consists of those graphs for which all cycles can b e covered with a small set of vertices
The second family consists of those graphs for which all cycles can b e covered with
a small set of edges In some limited sense these families are likethevertex cover
families studied in the previous chapter where a vertex cover can b e thoughtofasa
set of vertices that cover all edges This chapter also illustrates the use of testsets
to prove or disprove minimalityof t b oundaried graphs
Intro duction
The characterization of graph families based on the following twowell known prob
lems is the fo cus of this chapter see GJ
Problem FeedbackVertex Set FVS
Input A graph G V E and a p ositiveinteger k jV j
Question Is there a subset V V with jV j k suchthat V contains at least one
vertex from every cycle in G
Aset V in the ab ove problem is called a feedback vertex set for the graph G The
family of graphs that have a feedbackvertex set of size at most k is b e denoted by
k Feedback Vertex Set Itiseasytoverify that for each xed k the set of graphs
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
in k Feedback Vertex Set is a lower ideal in the minor order For a given graph
G let FV S G denote the least k such that G has a feedbackvertex set of cardinality
k Our second problem of interest is now stated
Problem Feedback Edge Set FES
Input A graph G V E and a p ositiveinteger k jE j
Question Is there a subset E E with jE j k suchthat G n E is acyclic
The edge set E is a feedback edge set Also for a given graph G let FES G
denote the least k suchthat G has a feedback edge set of cardinality k and the family
k Feedback Edge Set fG j FES G k g
Example Displayed b elowisagraphinthe Feedback Vertex Set fam
ily Notice that when the blackvertices are removed from the example the graph
b ecomes acyclic a forest
The reader should note that the graph in the previous example requires edges
in any feedback edge set and thus it is a member of Feedback Edge Set
One classic application for minimum feedback sets in directed graphs has to do
with op erating systems Consider a graph that mo dels pro cesses and system resources
as vertices and pro cessor requests or needs as edges When deadlo cks o ccur in the
system a feedbackedgesetcanbepicked to refuse selected pro cessor requests or a
feedbackvertex set can b e picked to shut down pro cessors or resources
For b oth typ es of these parameterized families a standard slow p olynomial time
algorithm exists for memb ership testing based on the brute force technique of check
n
ing all covering p ossibilities where n equals the number of vertices for FVS and k
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
k
numb er of edges for FES We used a somewhat practical O k n algorithm for
k Feedback Vertex Set while debugging our linear time dynamic program i e
our nite index congruence for t parses This algorithm byFellows and Downey see
DF is based on a quick algorithm by Itai and Ro deh in IR for nding short
cycles and the fact that a graph G of minimum degree three with girth at least k
is not in k Feedback Vertex Set For the feedback edge set problem weshowin
Section how toeasilycheck if a graph is a member of k Feedback Edge Set
The FVS Obstruction Set Computation
Wenow fo cus on two problem sp eci c details for nding the k Feedback Vertex Set
obstruction sets a nite index congruence and a complete testset i e steps and
of Section We rst present a practical linear time algorithm for the feedback
vertex set problem on graphs of b ounded pathwidth treewidth in t parse form This
general purp ose algorithm is altered to act as a nite index congruence that is a
re nement of the canonical congruence We then showhow to pro duce testsets for
the graph families k Feedback Vertex Set k with resp ect to any b oundary
size t
A nite state algorithm
Throughout the following discussion the b oundary size and width of a t parse is
xed Recall that the current set of b oundary vertices of a t parse G is denoted by
n
the symb ol For any subset S of the b oundary we de ne the following for all
pre xes G of G m n
m n
least k such that there is an FVS V of G with V S and jV j k
m
F S
m
otherwise whenever G n S contains a cycle
For any witness set V of G consisting of F S vertices there is an asso ciated
m m
witness forest consisting of the trees that contain at least one b oundary vertex in G n
m
V A witness forest tells us howtight the b oundary vertices are held together Some
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
of these forests are more concise than others for representing howvertex deletions
can break up the b oundary
For two witness forests A and B with resp ect to F S wesay A B if the
m w
following two conditions hold
For anytwo b oundary vertices i and j i and j are connected in A if and only
if i and j are connected in B
If for any t parse extension Z where there exists some non b oundary vertex b
of B suchthat B nfbg Z is acyclic then there exists a non b oundary vertex
a of A suchthat A nfag Z is acyclic
Also two witness forests A and B are equivalent A B if A B and B A
w w w
A witness forest in reduced form minimal number of vertices is called a park The
next lemmaprovides a way of cleaning up a forest to yield a park
Lemma A witness forest W of G may b e reduced to a park as follows
m
a all leafs end vertices not on the b oundary may b e pruned and
b any non b oundary vertex v of degree twomayhave an incidentedgecontracted
if the neighb orho o d N v
Pro of We rst show that any separable information is not lost after doing either
of the ab ove op erations
Let v b e a non b oundary end vertex of W Sincev is not on the b oundary of W
there do es not exist an extension Z suchthat W Z has a cycle containing v Vertex
v always has degree one Thus all end vertices of W not on the b oundary can not
b e included with the other witness vertices asso ciated with the witness forest W in
any minimal feedbackvertex set of any extended G
m
Now assume v is a non b oundary vertex of degree twoofW and N v fa bg
where a Let Z b e an extension of W such that the removal of vertex v kills
some cycles of W Z Since the degree of v is two all cycles through v must also
pass through a Thus vertex a is also a kill vertex for the cycles killed by v in W Z
This shows that wemay replace vertex v with vertex a in any feedbackvertex set
containing v Vertex v always has degree two
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
Let W b e the forest W with edge a v contracted The vertices v and a of
c
W are replaced with the vertex lab eled a in W Wenowshow W W For all
c c w
extensions Z there is a bijection b etween cycles in W Z and cycles in W Z All
c
cycles that pass through v of W Z now pass through a cycle with one less edge in
W Z all other cycles are iden tical For any cycle killed byvertex v in W Z the
c
corresp onding cycle in W Z is still killed byvertex a The other vertices of W or
c c
W still kill the same cycle extensions Thus W W and W W
c w w c
Wenowshow that the reduced park P derived from W using steps a and b
is minimal
Let vertex v b e a non b oundary vertex of P SinceP is acyclic and contains no
end vertices adjacent to the b oundary vertex v is on some unique path b etween two
b oundary vertices i and j Deleting v disconnects i and j So P nfv g P
w
Now let P b e the forest P where edge a b iscontracted for two non b oundary
vertices a and b of degree three or more Let a and a be two distinct b oundary
vertices connected to vertex a such that vertex b is not on the connecting paths
Likewise Let b and b be tw oboundaryvertices connected to vertex b such that
vertex a is not on the connecting paths The vertices a and a are distinct from the
vertices b and b for otherwise a cycle would contain edge a b in P Pick a graph
extension Z to b e the set of b oundary vertices S with the edges a a and b b
The graph P nfag Z is acyclic while the graph P Z contains twodisjoint cycles
This tells us that P nfxg Z is cyclic for all x in P n Thus P P
w
Finally assume P is the forest P where edge a b is contracted a b
and deg r ee b Let b and b be two b oundary vertices connected to vertex b
such that the path b etween b and b passes through vertex b and a fb b g Pick
an extension Z to b e the set of b oundary vertices S with the edges a b and a b
The graph P nfbg Z is acyclic The graph P Z contains two cycles whichintersects
at a Since the b oundary vertex a is not allowed to b e deleted the graph P nfxg Z
is cyclic for all x in P n Thus P P
w
There may exist alternative witness forests that preserve minimum sized feedback
vertex sets for all extensions of G A witness forest W is considered to b e a park if
m
the ab ove lemma can not b e applied to W
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
Lemma There are at most t vertices in any park for b oundary size t
Pro of First we consider the degree two non b oundary vertices For sucha vertex v
each of its neighb ors must b e a b oundary vertex After viewing v and its two incident
edges as a single edge b etween twoboundaryvertices we see that at most t such
vertices can o ccur Otherwise a cycle would exist on the b oundary
Nowwe consider the remaining non b oundary vertices Let p b e the number of
suchvertices and e b e the edge size of the subpark Using the fact that the size of
a forest must b e strictly less than the order wehave e t p Since the sum of
the vertex degrees is twice the size wealsohave t p e Combining these
inequalities while solving for p weget
t p
e t p or p t
Summing up the b oundary t the degree twovertices t and the degree
three or more vertices t shows that the order of any park can b e at most t
Corollary There is a nite numb er of parks with b oundary size t
Pro of Since wehave a b ound on the number of vertices for a park we can apply
Cayley s Tree Formula i e bycounting the numb er of lab eled trees forests to get
n
a b ound on the total numb er of distinct parks There are n lab eled trees of order
n
The result of the previous lemmamay b e strengthened to arrive at a tighter b ound
on the numb er of p ossible parks See for example the closely related Lemma
on page However this b ound is su cient for our purp oses that is to see that
there is a manageable constant number of parks This means that the following
algorithm can b e used as an usable nite index congruence
n S of the set of b oundary vertices For each subset S with complement S
our algorithm keeps track of the related parks in the following sets
P S fP j P is a park of G with leaves and branches over S g
m m
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
Nowwe nally present a linear time dynamic programming algorithm for the FVS
problem which is used as our nite F congruence for t parses This general purp ose
algorithm has the same structure as the vertex cover algorithm used in Chapter
indicating a standard approach for developing them The one pass algorithm simply
makes a transition from one state to another for each op erator of a t parse G
n
n n
t g g Thus after all the parks fP S j S g are determined
n m
for G all the parks fP S j S g for i m n are never referenced and may
m i
b e discarded
Our algorithm given in Figure starts by setting the sizes for the minimal
feedbackvertex sets on G G the edgeless graph with t b oundary vertices
m
This is done for all S There is only one park asso ciated with F S atthis
stage namely the isolated forest with t jS j vertices We break up the dynamic
step into cases dep ending on what typ e of op erator is at p osition m and the
condition selected in S or not of any a ected b oundary vertices of G or G
m m
These transitions are describ ed in cases of Figure When the algorithm reaches
the end of the t parse it computes the minimum number of vertices needed in any
feedbackvertex set for G by taking the least F S
n n
For space reasons weleave out the self evident rules required to up date the sets of
parks P S throughout each iteration of step I I of the FVS algorithm This pro cedure
i
essentially entails extending the parks with the current op erator and reducing them
by the rules given in Lemma and combining park sets if the two F s are equal
m
in cases
Example The following Table ofvalues for F S shows the application
m
of the FVS algorithm with the parse given in Example on page As can b een
seen by examining the graph in Example a minimum feedbackvertex set has
cardinality which corresp onds to the minimum value in the last column
n n
Theorem F or any t parse G t g g the algorithm in
n n
Figure correctly computes FV S G
n
Pro of For part I of the algorithm we note that jS j vertices are selected from the
b oundary of G for each S Thus the minimum feedbackvertex set for sucha
requirement is initially set that is F S jS j
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
I For m and every S set
F S jS j
II For m n do the following cases
n
Case vertex op erator i and i S
F S minfF S F S fig g
m m m
n
Case vertex op erator i and i S
F S minfF S F S nfig g
m m m
ij where i S or j S Case edge op erator
F S F S
m m
Case edge op erator ij where i S and j S
a If the edge op erator creates a cycle on S in G or F S then
m m
F S
m
b If there exists a park in P S suchthat i and j are in di erent trees then
m
F S F S
m m
else
F S F S
m m
III The minimum feedbackvertex set order of G is
minfF S j S g
n
Figure A general feedbackvertex set algorithm for t parses
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES n n n n ertex set state tables computed for Example kv eedbac able F g m m T g g g g g g g S f f f f f f f
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
n
For a vertex op erator i app ended to G we relab el vertex i of G as i and lab el
m m
the new vertex in G as i The correctness for the dynamic step of the algorithm
m
part I I is nowbeproved
Case F S minfF S F S fig g
m m m
n
First for the m th op erator b eing i where i S weshow that F S
m
minfF S F S fig g Let K b e a witness feedbackvertex set of G where
m m m
S K or S fig K The graph G resulting from adding an isolated
m
vertex to G with new b oundary vertex i also has K as a feedbackvertex set but
m
with K S
Nowwe show that minfF S F S fig g F S Assume K is a minimal
m m m
feedbackvertex set of G Vertex i is not in K since removing i from K would leave
m
a smaller feedbackvertex set for G contradicting K b eing minimal If i K
m
then K is a witness for G where K S fig Likewise if i K then K is a
m
witness for G where K S Thisthenshows that either F S F S or
m m m
F S fig F S
m m
Case F S minfF S F S nfig g
m m m
n
This case assumes that the next op erator is a vertex op erator i and i S We
S minfF S F S fig g Let K b e a witness feedback rst show F
m m m
vertex set of G where S K or S fig K Adding an isolated vertex to
m
G with new b oundary vertex i has K K fig as a feedbackvertex set for G
m m
with K S Thus the inequality holds this way
Now assume that K is a witness feedbackvertex set for the graph G and
m
S nfig g jK j minfF S F
m m
If i K then K K nfig is a witness for G where K S Likewise if i K
m
then K K nfig is a witness for G where K S nfig Thus we either have
m
F S F S or F S nfig F S
m m m m
Case F S F S
m m
ij to G where either vertex i The net result of adding an edge with op erator
m
or vertex j is marked for deletion is the same as if this op erator was not present The
edge gets deleted from the graph when the designated selected b oundary S is a subset
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
of the minimal feedbackvertex set with resp ect to F S of G IfF S F S
m m m m
then the witness feedbackvertex set for G would also b e a witness feedbackvertex
m
set for G i e a contradiction of F S b eing minimal
m m
Case F S F S orF S or
m m m
If the edge op erator ij creates a cycle on the non selected b oundary vertices
S n S or F S then there is no feedbackvertex set for G Thus
m m
F S is correctly set to
m
Wenow consider the cases where F S is nite Clearly F S F S
m m m
since G n S is a prop er subgraph of G n S
m m
Assume there is a park for F S such that b oundary vertices i and j are in
m
di erent trees This means that there exists a feedbackvertex set K of cardinality
F S of G with K S that disconnects the vertices i and j Adding an edge
m m
i j with op erator ij to G n K do es not create any cycles Thus G n K
m m
G f i j g n K G n K f i j g is acyclic In this case F S F S
m m m m
If the ab ove case is not true then all parks have the b oundary vertices i and
j connected Since F S there must b e at least one park witness forest
m
asso ciated with a minimal feedbackvertex set K of G Since op erator ij do es
m
not create a cycle on the non selected b oundary the unique cycle created in G n K
m
by adding the edge i j has at least one non b oundary kill vertex v Hence the set
K fv g is a feedbackvertex set of G Wehave shown F S F S
m m m
Wenow consider the p ossibilitythat F S F S in this latter case Let K
m m
b e a witness for G of cardinality F S The set K is also a feedbackvertex set
m m
for G Since K is a minimal feedbackvertex set for G the witness forest G n K
m m m
must havevertices i and j connected by assumption that no park disconnects i and
wever adding the edge i j to G n K causes a cycle This is a contradiction j Ho
m
since G n K G f i j gn K G n K f i j g Recall that i and j are
m m m
not in S So F S F S
m m
The dynamic program given in Figure for determining the feedbackvertex
set of a pathwidth t parse is easily mo di ed to handle treewidth t parses All that is
needed is to add a case in part I I whichtakes care of the circle plus op erator G G
i j
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
This new case is a little messy since the states for the two subtree parses G and G
i j
need to b e interweaved Brie y stated this is done bychecking all combinations
unions of b oundary subsets S and S of G and G resulting in a subset S of
i j i j
G G along with checking which b est parks from G can b e glued together with
i j i
the compatible parks from G to form a set of parks for G G If the glued parks
j i j
create any cycles then the value of F S needs to b e increased to account
tree index
for additional kill vertices
Using the same technique that we did for building our nite state algorithm
for k Vertex Cover we can convert the ab ove feedbackvertex set algorithm to
a nite index congruence for k Feedback Vertex Set This is accomplished by
restricting the values of F S tobeinf k k g we are only interested in
m
knowing whether or not there exists a feedbackvertex set of size at most k containing
S Thevalue of k acts as the value in the congruence
In our application for nding the k Feedback Vertex Set obstruction sets we
actually used a congruence with slightly fewer states then the one just describ ed The
key idea to this improvementwas noticing that if a park P is a minor of a park P then
only the representative P is needed as a witness We estimate that this allowed us
to proveapproximately more t parses nonminimal via the dynamic programming
congruence check That is for certain instances weavoided our testset pro of metho d
A complete FVS testset
A nite testset for the feedbackvertex set canonical congruence is easy to pro duce
F
The individual tests closely resemble the parks describ ed ab ove The testset that
we use consists of forests augmented with isolated triangles and or triangles solely
k
attached to a single b oundary vertex Our k Feedback Vertex Set testset T
t
consists of all t b oundaried graphs that have the following prop erties
Each graph is a member of k Feedback Vertex Set
Each graph is a forest with zero or more isolated triangles K s
Every tree comp onent has at least two b oundary vertices
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
Every isolated triangle has at most one b oundary vertex
Every degree one vertex is a b oundary vertex
Every non b oundary degree twovertex is adjacenttotwo b oundary vertices
k
Example Some b oundaried tests of T for Feedback Vertex Set
t
are shown b elow
k
The ab ove restrictions on members of T gives an upp er b ound on the number
t
k
of vertices as stated in the following lemma Hence T is a nite testset
t
k
Lemma The number of vertices for any test T T is at most k t
t
Pro of Since T k Feedback Vertex Set there can b e at most k isolated tri
angles consisting of at most k non b oundary vertices Wenow showby induction
that there can b e at most t interior forest vertices for b oundary size t With
out loss of generality wemay assume the acyclic part of T is a tree i e wecan
add edges to make another test with the same order For a tree with b oundary
vertices the largest test consists of one interior vertex of degree two Thus the base
case holds Now assume T is a valid test with t b oundary vertices We consider
three cases If T has a degree one b oundary vertex v that is adjacent to another
b oundary v ertex then T nfv g is a valid test for b oundary size t containing by
induction at most t interior vertices Hence T also has at most t t
interior vertices Otherwise if T has a degree twointerior vertex v then T nfv g
partitions the b oundary into twovalid tests T and T each with p ositive b oundary
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
sizes b b t By induction jV T j b b and jV T j b b so
jV T j t b b t Lastly if all of T s interior vertices have
degree at least then there must b e at least twice the number of leaves b oundary
vertices Thus any acyclic test T can haveatmostt interior vertices
The ab ove b ound is tight since the test T consisting of k isolated triangles and
t in terior degree twovertices each adjacent to b oundary vertex i and i has
k t vertices see for example the last test given in Example
Since these k Feedback Vertex Set testsets are based solely on t b oundaried
graphs they are useful for b oth pathwidth and treewidth t parse obstruction set
computations
k
Theorem The set of t b oundaried graphs T is a complete testset for the
t
graph family k Feedback Vertex Set
Pro of Assume G and H are two t b oundaried graphs that are not F congruent
for F k Feedback Vertex Set Without loss of generality letZ be any t
b oundaried graph that distinguishes G and H with G Z F and H Z F We
k
showhow to build a t b oundaried graph T T from Z that also distinguishes G
t
and H Let W b e a set of k witness vertices suchthat G Z n W is acyclic From
W let W W G W W and W W Z Take T to b e Z n W plus jW j
G Z Z
isolated triangles plus jW j triangles with eachcontaining a single b oundary vertex
from W IfT contains any comp onent C K without b oundary vertices replace
it with FV S C isolated triangles Clearly G T F since W plus one vertex
G
is a witness set of k vertices If from each of the non b oundary isolated triangles of T
H T F then this contradicts the fact that H Z F by using a cover containing
W W and the interior witness vertices of H with resp ect to H T Finally we
Z
k
construct a distinguisher T T by minimizing T to satisfy the prop erties listed
t
ab ove Note that the extension T is created by not eliminating any cycles in the
extension T
For the graph family Feedback Vertex Set on b oundary size the ab ove
testset consists of only tests However for Feedback Vertex Set on b ound
ary size the ab ove testset contains a whopping set of tests As can b e seen by
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
the increase in the numb er of tests a more compact feedbackvertex set testset would
b e needed if p ossible b efore we attempt to work with b oundary sizes larger than
The large numb er of tests esp ecially T for the FVS families indicates why using
the testset step to prove t parses minimal or nonminimal is the most CPU intensive
part of our obstruction set search andiswhy it is attempted last
The FVS Obstructions
Our search for the Feedback Vertex Set and Feedback Vertex Set ob
structions is now presented As mentioned in Chapter we need some typ e of
lemma that b ounds the search space The following well known treewidth b ound can
b e found in vL along with other intro ductory information concerning the minor
order and obstruction sets Weprovide a pro of in order to suggest how generous the
b ound probably is for the k Feedback Vertex Set obstructions whichisavery
small subset of the k Feedback Vertex Set family
Lemma Agraphink Feedback Vertex Set has treewidth at most k
Pro of Let G V E b e a member of k Feedback Vertex Set and V V be a
set of k witness vertices suchthat G G n V is acyclic The remaining forest G has
a tree decomp osition T of width Notice that a tree decomp osition T consisting of
the vertex sets of T augmented as T T V is a tree decomp osition for G of width
i
i
k
Corollary An obstruction for k Feedback Vertex Set has treewidth at
most k
Pro of Let G b e an obstruction and v anyvertex of G By de nition G G nfv g
k Feedback Vertex Set SinceG has a tree decomp osition T ofwidthatmost
k adding the vertex v to eachvertex set of T yields a tree decomp osition of width
at most k for G
k Feedback Vertex Set obstruc Wenow consider when the pathwidth of a
tion G can b e larger than the treewidth b ound of k If we attempt to build a path
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
decomp osition like the tree decomp ositions in the pro of of Lemma we see that
the forest G obtained by deleting an arbitrary vertex v and k witness vertices from
G hastohavepathwidth at least From EST we know that the forest contains
a sub divided K So such an obstruction must have at least k vertices For
pathwidth see Chapter or EST the forest has to contain one of the tree
obstructions of order and hence G has to have at least k vertices for
pathwidth to b e more than the treewidth plus one
Lemma If O is an obstruction to Feedback Vertex Set and has path
width greater than then O either has at least vertices or is also an obstruction
to k Pathwidth for some k
Pro of Assume that the pathwidth of O is andisnotapathwidth obstruction
If the pathwidth is larger than then wecangetalargervertex b ound There
must then exist a minor G of O with the same pathwidth as O Since O is a
Feedba ck Vertex Set obstruction the minor G must have a feedbackvertex
set V of cardinality If the forest G G n V has pathwidth or less we can build
a path decomp osition of G of width by adding the twovertices of V to the sets of
a path decomp osition of G of width So that leaves us with the case that G must
contain a tree of pathwidth at least Such a tree must have at least vertices so G
must have at least vertices Since O has the same pathwidth as G the obstruction
O of Feedback Vertex Set must also have vertices
Any connected obstruction to Feedback Vertex Set do es not contain dis
joint cycles or any degree one vertices or any consecutivedegreetwovertices so
having or more vertices seems unreasonable Observe that the graph K is an ob
athwidth not pathwidth struction to b oth Feedback Vertex Set and P
and that most of the k Pathwidth obstructions have p endentvertices and other
nonminimal prop erties so it is unlikely that the second case of the lemma is p ossible
Unfortunately at this time wehave not proven the imp ossibility of either of these
two cases We hop e with regards to Feedback Vertex Set thatwe can nd a
de nitive pro of and avoid a treewidth search
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
Besides the single obstruction K for the trivial family Feedback Vertex Set
the connected obstructions for Feedback Vertex Set and the connected obstruc
tions for Feedback Vertex Set pathwidth are shown in Figures
The two connected obstructions for Feedback Vertex Set were found in ab out
hours of accumulated CPU time when combining worker pro cesses a database
manager pro cess and a dispatcher pro cess running concurrently Our pathwidth
search for Feedback Vertex Set consumed over thousand hours of CPU time
running for ab out three months in duration while averaging workers from initially
a collection of SUN Sparcs and more recently including a few IBM s and
two Cray Y MPs
Table contains a brief summary of howmany pro ofs our system had to nd for
Feedback Vertex Set pathwidth The rst column states various starting
or restarting p oints in the search Lack of memory and disk space is the main
reason for the separate runs The second column gives the numb er of canonic non
b oundaried obstructions that havethegiven pre x The minimal no des column gives
the numb er of minimal t parse s that we encountered these are the in ternal no des of
our search tree plus any b oundaried obstructions The last column gives the total
numb er of graphs the system had to check This total includes those t parses that
were proved minimal or nonminimal or irrelevant The missing entries in the table
represent places that were fast dead end runs i e small subtrees of the search tree
leading only to nonminimal t parses and we did not b other keeping the pro ofs
We b elieve that Feedback Vertex Set may b e the only feasible family to
characterize since there are at least obstructions to Feedback Vertex Set
In fact this countisavery small p ercentage since we know of an obstruction with
order and wehave only searched through a subset of the graphs with maximum
order
In our display of the two obstruction sets for the within one twovertices of
acyclic families we present only the connected obstructions since any disconnected
obstruction O of the lower ideal k Feedback Vertex Set is a union of graphs from
S
k
O i Feedback Vertex Set such that FVS O k See Section
i
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
Table Summary of our Feedback Vertex Set obstruction set computation
pathwidth
Pathwidth Four Prefixes Obsts Extract Minimal Total
for Feedback Vertex Set File t parses proofs
x
x
x
x
x
K none
none
A
B
deg
C
D
E prefixed below
Prefix E E a
Prefix E E b
Prefix E E c
Prefix E E d
Prefix E E a
Prefix E E b
Prefix E E c
Prefix E E d
Prefix E E a E
Prefix E E b none
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
Example Since K is an obstruction for Feedback Vertex Set and K
is an obstruction for Feedback Vertex Set the graph K K is an obstruction
for Feedback Vertex Set
Some patterns b ecome apparent in these two sets of obstructions suchasthe
following easily proven observation
Observation For the family k Feedback Vertex Set the complete graph
K the augmented complete graph A K which has vertices f k g
k k
fv j i j k g and edges
i j
f i j j i j k g
f i v and v j j i j k g
i j i j
and the augmented cycle A C are obstructions
k
The FES Obstruction Set Computation
Wenow fo cus on two problem sp eci c areas for computing the k Feedback Edge Set
obstruction sets a direct minimality test and a complete testset i e steps and
of Section
A direct nonminimal FES test
We rst describ e a simple graph theoretical characterization for the graphs that are
within a few edges of acyclic This trivial result also shows that Problem i e
determining the minimum feedback edge set of a graph has a linear time decision
algorithm
Theorem A graph G V E with c comp onents has FES G k if and
only if jE j jV j c k
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
Pro of For k the result follows from the standard result for characterizing forests
If FES G k then deleting the k witness edges pro duces an acyclic graph and thus
jE j jV j c k Now consider a graph G with jV j c k edges for some k
Since G has more edges than a forest can have there exists an edge e on a cycle Let
G V E nfeg By induction FES G k Adding the edge e to a witness
edge set E for G shows that FES G k
Unlike the k Feedback Vertex Set lower ideals it is not obvious that the
family k Feedback Edge Set isalow er ideal in the minor order However with
the ab ove theorem one can easily prove this
Corollary For each k the family of graphs k Feedback Edge Set is a
lower ideal in the minor order
Pro of Weshow that the three basic minor op erations do not increase the number
of edges required to remove all cycles of a graph An isolated vertex deletion removes
bothavertex and a comp onent at the same time so k is preserved in the formula
jE j jV j c k For an edge deletion the number of components can increase by
at most one so with jE j decreasing by one the value of k do es not increase For
an edge contraction the number of vertices decreases by one the numb er of edges
decrease by at least one and the numb er of comp onents stays the same so k do es
not increase
The ab ove corollary allows us to characterize each k Feedback Edge Set lower
haracterize these b elow ideal in terms of obstruction sets We abstractly c
Theorem A connected graph G is an obstruction for k Feedback Edge Set
if and only if FES G k andevery edge contraction of G removes at least two
edges i e the op en neighb orho o ds of adjacentvertices overlap
Pro of This follows from the fact that an edge contraction that do es not removeat
least two edges is the only basic minor op eration that do es not decrease the number
of edges required to kill all cycles for a connected graph with every edge on some
cycle
The ab ove theorem gives us a precise means of testing for nonminimal t parses see step of Section
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
A complete FES testset
Somewhat surprisingly a usable testset for each feedback edge set family has already
b een presented in Section Wenow prove that those earlier feedbackvertex set
tests can also b e used here
k
Lemma The testset T for the family k Feedback Vertex Set is also a
t
testset for k Feedback Edge Set
Pro of First observethat
F k Feedback Edge Set k Feedback Vertex Set
k
so that the k Feedback Vertex Set memb ership restriction for T graphs do es not
t
preclude any imp ortant tests just includes some obsolete tests not in F Consider
a xed family F and b oundary size t Itsu cestoshowthatifG H then there
F
k
exists a test T T that distinguishes G and H SinceG and H are not congruent
t
there exists a t b oundaried graph Z such that without loss of generality G Z F
k
and H Z F Wenowshowhowtominimize Z into a T T LetE b e a witness
t
edge set for G Z F and let E E Z n E The rst transformation on Z is to set
Z
Z Z n E jE j K Clearly Z is also a distinguisher for G and H since
Z Z
G Z F by using the edges E n E and one edge from each of the new K s as a
Z
is witness set and H Z F for otherwise H Z would b e in F Notice that Z
a set of trees and isolated triangles The nal transformation on Z is to let Z be Z
with all non b oundary leaves deleted and non b oundary sub divided edges contracted
k
to satisfy the conditions of a member of T
t
It is interesting to notice from the ab ove pro of that in addition to the out
of family tests the isolated triangles in the tests for k Feedback Edge Set do
not contain any b oundary vertices Thus the numb er of graphs in a testset for
k Feedback Edge Set is substantially smaller than the order of the testset for
k Feedback Vertex Set
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
The FES Obstructions
Since the family k Feedback Edge Set is contained in k Feedback Vertex Set
the maximum treewidth of any obstruction for k Feedback Edge Set is at most
k Thus the same arguments given in Section regarding pathwidth apply to
k Feedback Edge Set as well
For the family Feedback Edge Set it is trivial to showthatK is the only
obstruction The connected obstructions for the graph families Feedback Edge Set
through Feedback Edge Set are shown in Figures and There are
well over connected obstructions for the Feedback Edge Set family Any
disconnected obstruction for k Feedback Edge Set is easily determined by com
bining connected obstructions for j Feedback Edge Set j k since FES G
FES G FES G G
An op en problem is to determine a constructive metho d for nding all of the
obstructions for k Feedback Edge Set directly from O j Feedback Edge Set
j k Some easily observed partial results are given next
Observation If G is a connected obstruction for k Feedback Edge Set
then the following are all connected obstructions for k Feedback Edge Set
G with an added sub divided edge attachedtoanedgeof G
G with an attached K on one of the vertices of G
G with an added edge u v when there exists a path of length at least twobetween
u and v in G n E for each feedback edge set E of k vertices
It is easy to see that if an obstruction has a vertex of degree two then it is pre
dictable byobservations The rst Feedback Edge Set obstruction in Fig
ure wheel W and the second Feedback Edge Set obstruction in Figure
W aretwo examples of graphs where observation predicts the graph Those
Feedback Edge Set and Feedback Edge Set obstructions pathwidth b ound
ertices and cut vertices are shown in Figures and of without degree twov
The third Feedback Edge Set obstruction in Figure is not predicatable from
CHAPTER FEEDBACK VERTEX EDGE SETS FVS AND FES
the Feedback Edge Set obstructions by using any of the ab ove observations
Here deleting any edge from this obstruction leaves a contractable edge that do es
not removeany cycles that is all single edge deleted minors are nonminimal see Theorem
Figure Connected obstructions for Feedback Edge Set
K A K A C
Figure Connected obstructions for Feedback Vertex Set
Figure Connected obstructions for Feedback Edge Set
K
A C A K
Figure Connected obstructions for Feedback Vertex Set pathwidth
Figure Known connected obstructions for Feedback Edge Set
Figure Biconnected Feedback Edge Set obstructions without degree ver
tices pathwidth
Figure Biconnected Feedback Edge Set obstructions without degree ver
tices pathwidth
Chapter
Some Generalized VC and FVS
Graph Families
This chapter develops problem sp eci c results leading to the characterization in
terms of obstruction sets for several parameterized graph families The previous
two k Vertex Cover and k Feedback Vertex Set graph families are general
ized into other graph covering families that are lower ideals in the minor order We
also include e cient dynamic programs and computer generated automata examples
for various a path and cycle cover problems where the input has b ounded combina
torial width
Path Covers
Wenow generalize the k Vertex Cover graph families see Chapter These
path cover families have applications in the design of minimal broadcast graphs via
graph comp ounding see BFP
De nition For a graph G letM axP ath G b e the length numb er of edges
of the longest path The base level graph family is
Path Cover p fG j M axP ath G pg
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
The graph family k Path Cover p consists of the graphs that are within k vertices
of Path Cover p that is a graph G V E isink Path Cover p if there
exists a V V jV j k suchthatM axP ath G n V p
Note that the parameterized family of graphs k Path Cover is identical to
the set of graphs in the k Vertex Cover family
Lemma The graph family k Path Cover p is a lower ideal in the minor
order
Pro of Let G V E b e a graph in k Path Cover p with V V acovering
For any edge e u v E letG G nfeg If either u or v is in V then
G n V G n V has a maximum path of length at most p Ifnot G n V G n V and
its maximum path length is less than or equal to p Now consider an edge contracted
minor G G e Ifeitheru or v is in V then G n V G n V has a maximum path
of length at most p If not G n V is an edge contracted minor of G n V and edge
contractions do not increase the lengths of any paths So M axP ath G n V p
The following result follows from the treewidth result given in FL b but is
included and sp ecialized here for completeness
Theorem The pathwidth of any graph is at most the height of its smallest
depth rst search DFS spanning tree
Pro of Consider a DFS spanning tree T with ro ot r of a graph G Letv v v
m
b e the leaves of the tree T in visit order Let V b e the set of vertices from G on the
i
path from r to v in T We claim that V V V is a path decomp osition of G
i m
The width of this decomp osition is the height of the tree T Since a DFS spanning
tree do es not haveany cross edges anyedge u v G has b oth vertices in some
V Now consider i k j m Supp ose there is a vertex v G suc h that
i
v V V Vertex v is not a leaf of T since each leaf is in exactly one V Also
i j i
for any non leaf vertex v v V implies that v was visited b efore v in the DFS tree
i i
Let v b e the nearest vertex to v in V V of T Vertex v must b e on the path from
i j i i j
the ro ot r to v to b e in b oth V and V Notethatv comes after v in the DFS
i j i j i k i j
visit order This implies that the path r v v is a subpath of the path
i i j
r v So v V and V V V
i k k i j k
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
Corollary The maximum pathwidth of any graph G in k Path Cover p is
k p
Pro of Let W b e a set of witness vertices suchthat M axP ath G n W p Every
comp onent C of G n W has a DFS spanning tree of height at most p Thus combining
i
the vertex set W with the path decomp ositions of width at most p for each comp onent
C shows that G has pathwidth at most k p
i
Notice that the complete graph K has pathwidth k p and is a member of
k p
k Path Cover p However wedonotbelieve the next result is tight
Corollary If G is an obstruction for k Path Cover p then the pathwidth
of G is at most k p
Pro of The minor G Gnfv g for anyvertex v G is a member of k Path Cover p
Thus G has pathwidth at most k p and G has pathwidth at most k p
A path cover congruence
Wenow consider the family Path Cover p A dynamic programming congruence
for t parses is given b elow Later we indicate how to handle the within k vertices cases
Our nite index congruence for Path Cover p is based on a general purp ose
M axP ath algorithm The states of this dynamic program consists of lengths
indexed by path sequences of the form
I b P jG b P jG b b P jG b I
s s
where b b b are distinct b oundary vertices s t
s
We use square brackets to denote optional b eginning and trailing I s The nota
tion P jG means that either P or G is that letter of the sequence The semantics of
the variables I P and G are
The variable I denotes a path to from an interior vertex and the b oundary
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
The variable P denotes a path b etween two b oundary vertices this can b e a
b oundary edge
The variable G denotes a gap in the path contributes to the length
Each length is set to if no such path is p ossible otherwise it is the maximum
sum over all disjoint partial paths b etween an interior vertex and b allb and b and
i i
b and another interior vertex where applicable Note that the degenerate sequence
s
II denotes if p ositive the length of a maximum path in the interior
We also ignore path sequences that have the substring Gb G since replacing
i
that substring with G represents the same typ e of future path i e two gaps of
length add up to
Lemma For b oundary size t there are a nite numb er of path sequences
Pro of A simple upp er b ound is obtained bycounting the legal combinations of the
variables There are ways that the variable I can app ear e g one or zero times at
the front or end of the sequence If there are j distinct b oundary vertices then
j
there are p ossible ways the variables P and G are inserted b etween consecutive
b and b overcounting for consecutive G s The remaining one b oundary vertex
i i
case has t p ossibilities After combining these choices wehaveatmost
t
X
t j
A
t t t t t t j
j
p ossible path sequences
We note that approximately one half of the path sequences counted in the
previous lemma need to b e kept since a sequence s s s and its reverse
m
s s s representthesametyp e of path
m m
Example The path sequence indices of interest i e those path sequences
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
that determine a t parse s state for b oundary size are listed b elow
I I G I I
I I G I I
II I P
P I P
G I G I
I P I
The ab ove example illustrates that our b ound given in Lemma is not very
tight i e we need only path sequences verses the predicted by the lemma
Theorem There exists a linear time dynamic program that determines the
maximum path length of a t parse G
n
Pro of It su ces to showhow to correctly up date the ab ove path sequence state
table in constant time for b oth vertex and edge op erators The maximum length
over all non G path sequence indices is the maximum path of the t parse Initially
n n n
for the empty pre x t parse G t all sequence lengths are set to
except that the following lengths havevalue
b Gb and b where b b and b b
ij Case edge op erator
For this simple case we just need to up date only those indices for which adding
an edge b etween b oundary vertex i and j can create a longer path That is if the
length m of a path sequence iG j is not then the length of iP j
is at least m Here we set the length for this sequence to the maximum of its
previous value and m By de nition no other path sequence needs to b e increased
For example the length for the trivial case iP j where i and j are disconnected
b oundary vertices would b e set to b ecause iG j always has length
n
Case vertex op erator i
For this case we need to incorp orate maximum length paths that go through
the b oundary at vertex i into paths that go through the interior Wesay that a
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
k
i i i
j j
Snag case
Middle of path case
One end case
I iP j I iI kP iP j
Ij II kP j
Rare middle case
kGiP j kGj I
Figure The vertex op erator sub cases for the pro of of Theorem
new b oundary vertex i replaces the old b oundary vertex i The p ossible cases to
consider the symmetric representative cases are displayed in Figure A picture
aid of what path sequences to p ossibly alter is also given in the gure Since none
of the physical paths of the pre x t parse G are increased in G our dynamic
m m
program simply recategorizes some of the path sequence lengths Again likethe
edge op erator case the maximum of the previous sequence length and the length
of the new candidate path is used to determine a path sequence s new length The
old path sequence containing b oundary vertex i on the path may b e assigned the
i value since vertex i is now isolated The path sequences with a gap G between
and another b oundary vertex j need not b e considered in this up date pro cess For
example during the moveofboundaryvertex i to the interior the path sequence
IiGj represent a path for j but j already represents another of
path with greater or equal length
For b oth the ab ovetwo op erator cases the up date pro cess from G to G is
m m
done in time b ounded by the constantnumb er of path sequences Since this happ ens
only n times for the t parse G theoverall running time is linear
n
Corollary For a maximum path length p i e for the family Path Cover p
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
our M axP ath algorithm can b e used as a nite index congruence
Pro of This is accomplished by b ounding the lengths of each path sequence to p
Since the maximum length over all of the path sequences never decreases any
extended t parse stays out of the Path Cover p family Ifx is the total number
of path sequences the numb er of equivalence classes of this congruence is b ounded
x
by p
We can convert the nite state algorithm given in the pro of of Corollary to
recognize graphs that are within k vertices of having p as the maximum path length
i e a k Path Cover p dynamic programming congruence This is accomplished
by using the same technique that we used for our nite state k Vertex Cover and
n
k Feedback Vertex Set algorithms That is we break the two op erator cases i
and ij down into sub cases that dep end on conditions for each p ossible kill set S
and keep witnesses for each of the p ossible killed vertices in the interior
From the previous dynamic program we also get the following simple corollary
Corollary For graphs of b ounded pathwidth i e t parses the Hamiltonian
path problem can b e solved in linear time
Pro of We just run the linear time M axP ath algorithm on a t parse G and check
if the answer equals jGj
Cycle Covers
Wenow generalize the k Feedback Vertex Set graph families see Chapter
De nition For a graph G letM axC y cl e G b e the length of the longest
cycle The base level graph family is
Cycle Cover l fG j M axC y cl e G l g
The graph family k Cycle Cover l consists of the graphs that are within k vertices
V E isink Cycle Cover p if there of Cycle Cover l that is a graph G
V jV j k suchthatMaxCycle G n V l exists a V
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
The cycle cover family k Cycle Cover is identical to the set of graphs in
k Feedback Vertex Set The two degenerate families k Cycle Cover and
k Cycle Cover are classi ed as follows the rst family
k Cycle Cover fG j G has at most k verticesg
has the single obstruction k K consisting of isolated vertices and the second
family
k Cycle Cover k Vertex Cover k Path Cover
has already b een seen
Lemma The graph family k Cycle Cover l isalower ideal in the minor
order
Pro of This pro of is similar to the pro of given for Lemma except that we observe
that taking either subgraphs or edge contractions of a graph do not increase the length
of any cycle
The next result relating treewidth and the maxim um cycle length was rst proved
byFellows and Langston FL b They show that if a graph has maximum cycle of
length k noback edge in any DFS tree can go back more than k levels One
then reads o a tree decomp osition of width k fromsuch a DFS tree Our pro of
below is recursive and is based on separator sets and illustrates the fact that each
biconnected comp onent of a graph with a maximum cycle of length k canonlyhave
another disjoint cycle of length at most k
Theorem For any graph G tr eew idth G M axC y cl e G The degener
ate cases are MaxCycle K and for any forest F with at least one edge
MaxCycle F
Pro of We rst note that the theorem holds for the degenerate cases since clearly
K has treewidth and any forest has treewidth at most Weprove the other cases
by induction on the maximum cycle length For a base case consider a graph G
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
with M axC y cl e G Since G do es not contain any cycles of length G do es
not contain either K or K as a minor Thus G must b e outer planar From
Lemma we know that G has treewidth at most and so the base case holds
Now consider a graph G with MaxCycle G k k Without loss of gen
erality we can assume G is biconnected If G is not biconnected then a set of
minimum tree decomp ositions T T T of the maximal biconnected subgraphs
m
S S S of G can b e pasted together to form a tree decomp osition T of width
m
equal to the maximum width of any T Let C b e a maximum cycle in G and
i
C C C m b e the connected comp onents of G n C Ifm thenthe
m
be the vertex induced subgraph treewidth pathwidth of G is at most jC j Let C
i
of G with vertices in the closed neighborhood of C N V C Also let p bethe
i i i
number of vertices in C adjacenttoC thatis p jV C V C j We build a tree
i i
i
decomp osition of G with ro ot vertex T C combined with child tree decomp ositions
r
and show that the resulting tree decomp osition has width from the comp onents C
i
at most k
We rst take care of any comp onent C with just one vertex v The graph C
i
i
which consists of vertex v and its neighbors on C can have at most bk c vertices
or otherwise C would not b e a maximum cycle in G Thus for any such comp onent
C we can attachavertex set consisting of V C to T
i r
i
Tomake the rest of the pro of more understandable we rst investigate the case
where p and then later generalize for each p Let v and v denote the
i i
twovertices in C C Consider a maximum cycle D jD j in C Since G is
i
i
d and P d v biconnected there are twovertex disjoint paths P v
between C and D d d on D as shown in Figure We claim that jD j k
k Let C b e the longest path b etween v and v in C D b e the longest path
p p
between d and d in D andE b e the cycle P D P C Wehave the following
p p
inequality
k k k k k k
k jE j jP j jP j
of the maximum cycle D in C is at most k By applying induction So the length k
i
wehave a tree decomp osition T of width at most k for C Ifwe add the p
i i i
vertices v and v to eachvertex set in T we get a tree decomp osition T of width at
i i
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
C G
D C
i
v
P
d
d
P
v
Figure Illustrating the p case for the pro of of Theorem
i
most k forC This subtree decomp osition T can b e attached byany connecting
i i
edge to the separating vertex set T
r
Finally we consider the generalized case p bk c withjC j Let
i i
v v v be the vertices in W C C Consider a maximum cycle D jD j
p
i
i
in C We partition the vertices in W into equivalence classes de ned by v v if
i i j
tpaths P v d and P v d and only if there do es not exists disjoin
i i i j j j
between C and D d d on D Since G is biconnected there are at least two
i
equivalence classes We use the same construction as in the p case given ab ove
i
and show that jD j k k p Assume v is a memb er of an equivalence class
i i
with m p memb ers Since anytwovertices v and v in W can not b e adjacent
i i j
on C the m vertices in v s equivalence class must preclude at least m p ositions
i
m vertices in W Likewise the other on C for the lo cations of the remaining p
i
equivalence classes collectively prevent at least p m p ositions on C This
i
means that there exists two non equivalentvertices v and v that have distance at
i j
most b k p m m c on C In other words v and v are connected
i i j
by a path of length at least k k p k p onC Let C be
i i p
this longest path b etween v and v in C D b e the longest path b etween d and d
j i p i j
in D andE b e the cycle P D P C Wenowhave the following inequality
i p j p
k jE j jP j jP j k p k
i
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
k p k
i
k k p
i
So as desired the length k of the maximum cycle D in C is at most k p which
i i
is less than or equal to k p By applying induction wehave a tree decomp osition
i
T of width at most k p forC Ifwe add the p vertices W to eachvertex
i i i i
set in T we get a tree decomp osition T of width at most k p for C This
i i
i i
sub tree decomp osition T can b e attached as we did for the p case to the
i
i
separating vertex set T
r
Corollary The maximum treewidth of any graph G in k Cycle Cover l
is k l
Pro of Let W b e a set of witness vertices such that M axC y cl e G n W l From
the ab ove theorem every comp onent C of G n W has treewidth at most l Thus
i
combining the vertex set W with the tree decomp ositions of each C and adding
i
arbitrary edges in a tree like fashion b etween the sub decomp ositions shows that G
has treewidth at most k l
Again as wesaw for the k Path Cover p families the complete graph K
k l
in k Cycle Cover l has treewidth k l We susp ect that the next result can
b e improved
Corollary If G is an obstruction for k Cycle Cover l then the treewidth
of G is at most k l
Pro of The minor G G nfv g for anyvertex v G is a memb er of the lower ideal
k Cycle Cover l Thus G has treewidth at most k l and G has treewidth
at most k l
Regarding memb ership algorithms there is an e cient linear time algorithm for
checking if a graph has a cycle of length at least and for checking k Cycle Cover
k
memb ership in time O n These algorithms are based on the following fact
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
Lemma A graph has a cycle of length at least if and only if it has a bicon
nected comp onent of at least vertices
Pro of If a graph has a cycle then clearly an induced biconnected comp onent con
taining the cycle has at least vertices Let B b e a biconnected comp onent with at
least vertices Since B is not a tree let vertices a b and c b e a triangle Without
loss of generality let vertex d fa b cg b e adjacenttoa Now since a is not a
cut vertex there exists a path from vertex d to either b or c avoiding vertex a But
this means that the four vertices fa b c dg are in some common cycle of length at
least
The ab ove fact do es not generalized for any cycle lengths greater than e g
k Cycle Cover since the graph with vertices in Figure is not Hamiltonian
Figure The smallest biconnected non Hamiltonian graph
With the recentpowerful result of Courcelle see CM Cou Cou b con
cerning nd order monadic logic of graphs we know for any xed k there exists a lin
ear time algorithm for Cycle Cover k that is this problem is xed parameter
tractable see DF Our application of this p owerful result is sketch next First
in the pro cess of nding a depth rst spanning tree T of a graph G if a back edge
which is adjacent to a parent no de creates a cycle of length greater than k then we
know the answer is no and can stop Otherwise with this DFS tree T we read o
a tree decomp osition of width at most k forG FL b The nd order monadic
logic representation of this problem tells us that there is a linear time algorithm for
graphs of b ounded treewidth For example consider this simple logic for k on a
graph G V E
v V v V v V v v v v
v v E v v E v v E
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
Path Cycle Cover Testsets
Recall that a testset for a canonical family congruence is a set T of b oundaried
graphs such that there exists a distinguisher Z in T for every two non congruent
graphs X and Y That is if X Y then either X Z F while Y Z F or vice
F
versa In this section we present testsets for various path and cycle cover families
With these testsets one can build practical memb ership automata for t parses
Some maximum path cycle testsets
We rst present testsets for the
MaxCycle l Cycle Cover l and
MaxPath p Path Cover p
graph families for t b oundaried graphs The MaxCycle l tests resemble the Hamil
tonian cycle tests presented in Section also see Lu Let F denote either
MaxCycle l or MaxPath p where l p Assume X and Y are b oundaried
graphs that are not congruent Without loss of generality let Z be a test such that
X Z F and Y Z F Consider an out of family path cycle P in Y Z and
let Z P Z Since Z Z wehave X Z F and Y Z F The test
Z has comp onents consisting of paths P P P where the b oundary of Z lies
m
on the end p oints of certain P In the case of MaxPath p at most two end p oints
i
l every end p ointofZ of Z are not b oundary vertices In the case of MaxCycle
is a b oundary vertex The ab ove discussion leads to the following result for any xed
b oundary size
Lemma A nite testset for MaxCycle l consists of tests of classes T and
T and a testset for MaxPath p consists of additional tests of classes T and T
a b c
see Figure
For illustration this testset mo dulo b oundary p ermutations for MaxPath p
is shown in Figure for b oundary size The testset for the MaxCycle l family
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
T
T
a
i
i p
j
P P
p p
i j p i
T T
b c
i i i
i p
j j
j
k
P P P P
p p p p
i j k p
j p i i j p i i
Figure Four typ es of b oundaried tests for the MaxPath p and MaxCycle l
graph families l p
consists of only tests of typ e T and T Thevariables i j and k denote lengths of
a
the paths and cycles of the tests The summations and inequalities indicate how
many tests in each class
A maximum path automaton example
Using the MaxPath p or MaxCycle l testsets given ab ove we can build fast
memb ership automata that recognize t parses in those graph families A straightfor
ward way of building automata from testsets is discussed in Section Further
information regarding building automata for b ounded pathwidth and treewidth prob
lems o ccur in the literature e g see AF APS KK b
For a simple example consider the family MaxPath over parses In Ta
ble wegive a transition diagram for the minimal nite state automaton that
recognizes those graphs of pathwidth caterpillars that have no paths of length
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
Table The transition diagram for the MaxPath automaton
parse op erators reference counts
n n
State during O set search
or greater State of this automaton is the out of family state A pictorial view of
this automaton is given in Figure
Also given in the right three columns of Table for the MaxPath au
tomaton is a reference count for eachcombination of state and alphab et symbol
transition during the search for the single obstruction P path of length See
Section for more details The zero entries showthatmanybranches where not
taken during this typ e of obstruction set search This indicates that computing ob
struction sets via automata is not very e cientoverall In practice we found that
building an automaton takes over of the computation time while searching for
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
start
Figure The automaton for MaxPath corresp onding to Table
obstructions using a minimum automaton is very fast by the search technique of
Chapter Our empirical results suggest that for obstruction set computations if
one has an automaton or minimal congruence available use it otherwise consider
another metho d
Some generic cycle cover testsets
Wenow describ e how to create a testset for the graph family k Cycle Cover l
using b oundary size t The tests are a combination of the k Feedback Vertex Set
tests see Section and the maximum cycle tests given ab ove We illustrate the
testset construction with the k Cycle Cover family
k
Our k Cycle Cover testset consists of all t b oundaried graphs that have
t
the following prop erties
Each graph is a member of k Cycle Cover
All degree zero and one vertices are b oundary vertices There are no cycles of length greater than
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
All cycles C are isolated or attached to at most b oundary vertex
Any degree twovertex is on a path of length at most b etween some two
b oundary vertices
Anyedge e on an interior cycle C is allowed if there exists a path b etween
two b oundary vertices a and b containing e that is larger than any other path
between a and b
k
Lemma The set of t b oundaried graphs is a testset for k Cycle Cover
t
Pro of Let F k Cycle Cover for some xed constant k The pro of that
k
is a testset follows the same line of reasoning as our pro of of correctness for the
t
k Feedback Vertex Set testset For two graphs G and H such that G H
F
k
we need to show that contains a distinguishing test First we can assume b oth
t
G and H are members of F since the empty t b oundaried graph is a distinguisher
Without loss of generality let T be any t b oundaried graph suchthatG T F
and H T F It is safe to assume that the graph T do es not haveany degree zero
or one internal vertices b ecause anysuchvertex do es not contributed to any cycle
created by the op erator
If T contains anyinternal cycles of length four or greater then any kill witness set
W in either case must contain one vertex on these cycles So another distinguishing
test T created by adding j T W n j isolated C s to a pruned T W n where W
is a witness for G T F A graph is considered to b e pruned if it satis es prop ert y
k
of Now consider any remaining cycle C of length at least in T This cycle
t
must b e covered by a b oundary vertex of W This testing b oundary internal cycle
k
can b e replaced with a test that satis es prop erty of For the subset S W
t
we construct a test T that is a pruned T n S with jS j four cycles added each
containing one of the removed b oundary vertices in S Thus G T F while still
H T F IfH T F then wecan ndakillsetW containing T W where
jW j k that contradicts H T F
overtex that Prop erty is trivially seen since contracting out any such degree tw
do es not satis es this prop erty pro duces a smaller distinguisher For the validityof
the prop erty restriction we rst note that in order for an edge e u v to o ccur
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
in a T typ e test i e a test without cycles of length or greater the longer paths
must all detour around a neighbor vertex z on a cycle C fx y z g See test a in
Figure In this case removing the edge do es not allowforany smaller kill set for
the out of family H T This is b ecause a kill vertex z could have b een replaced
with x or y in a witness set
k
Note that there may b e other test reductions b esides the ones listed ab ovefor
t
suchascontracting edge e of Figure However wehave stated enough restrictions
to guarantee a nite testset
k
Lemma The cycle cover testset has nite cardinality
t
Pro of Toshow niteness it su ces to proveabound f t onthenumber of internal
vertices excluding C s that a connected test can have for b oundary size t Since a
test is a member of k Cycle Cover atmostk isolated four cycles can b e
added for a general test For our base cases f for the empty test and f
via a path of length
Wenow consider some structural p ossibilities for a test Notice that all vertices
of anyinternal C have degree greater than twosuch as the triangle fx y z g in
Figure b ecause of prop erty If the test has a C fx y z g then removing
the internal incident edges f x y x z y z g creates a comp onent test not
k
necessarily in If there were fewer than comp onents then wewould contradict
t
the assumption that there exists no cycles of length greater than To satisfy the
k
we can contract at most one edge incidenttoeachoffx y z g as prop erties of
t
shown in Figure Note that a contraction is not needed for a b oundary vertex
within the original C Since each of these smaller comp onents have at most t
b oundary vertices we get the following recurrence
t
f t f t
Now consider the case that a test has no C s in fact no cycles Following the
argument of Lemma mo di ed for these tests we see that if there is a degree two
internal vertex then f t f t Otherwise wemust have a tree of b ounded
height with all internal vertices of degree at least three Again the previous b ound
t
f t holds
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
x
z
e
x
y
x piece
e
y
y piece
z
z piece
a Edge e is not needed b Recursive pieces of previous test
k
Figure A t b oundaried test example in for k Cycle Cover
t
Other VC FVS Generalizations
The minimum c clique hyperedge cover e g vertex covers corresp ond to K binary
edge covers is de ned to b e the minimum number of vertices needed to cover all
cliques of size c of a graph A parameterized graph family based on this invariantis
k Clique Cover c fG j there exists a V V G with jV j k
suchthat G n V has no cliques of order c g
Unfortunately the graph family k Clique Cover c is not a lower order in the
minor order since graphs with a b ounded maximum clique size isnotalower ideal
and hence no obstruction set characterizations are p ossible see Lemma
The minimum c xed cycle cover feedbackvertex sets corresp ond to covering all
cycles is de ned to b e the minimum number of vertices needed to cover all cycles
of length exactly c The graph family with covers of size at most k is denoted by
k Fixed Cycle Cover c Likeabove the graph family k Fixed Cycle Cover c
is not a lower ideal in the minor order for c since the graph consisting of k
disjoint C cycles is in k Fixed Cycle Cover c but the minor consisting of one
c
edge contracted from each C is not
c
Note that the families k Clique Cover and k Fixed Cycle Cover are
identical for an y k since b oth require coverings of all K subgraphs For illustration
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
purp oses wenow present a nite state dynamic program for k Clique Cover
k Fixed Cycle Cover over t parses This algorithm designed in the same spirit
as our k Vertex Cover and k Feedback Vertex Set algorithms is based on a
general purp ose linear time algorithm that computes the size of any minimum sized
K cover
For this K cover dynamic program the state of a t parse G g g g at
n n
pre x p osition m n consists of these three comp onents
Induced b oundary subgraph of G
m
For each S
M S fC j C is the minimum K cover containing S g
m
For each S and for each witness cover C S of G with M S vertices
m m
u and v and there exists a w G n C
m
P C u v
m
suchthat u w and v w are edges in G n C
m
each P C has n C as an available b oundary attribute and
m
W S fP C j C S is a K cover for G of size M S g
m m m m
Note that it is not necessary to keep P C in W S if there exists another
m m
witness cover C for S suchthat C C and P C P C This is b ecause
m m
the following implication holds for any t parse extension Z
G Z n C k Clique Cover G Z n C k Clique Cover
m m
increases up to The following algorithm up dates these state comp onents as m
n At the end of the algorithm the minimum sized K cover of G is the value of
n
M Toconvert this algorithm to a xed k version for k Clique Cover we
n
simply place an upp er b ound of k onthevalues of the M S this tec hnique
m
of trimming also yields a nite state k Clique Cover algorithm since in this
i
P
j j
j j
di erent trimmedversion there are at most k values of M S and
m
i
i
P C m
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
Algorithm Finds minimum K cover of a t parse G g g g
n n
n n
Initial state for pre x G t
t
For all S
M S jS j
t
W S fP S g
t t
end
For m t n do for all S
n
Case g i and i S
m
M S M S nfig
m m
W S pullo W S nfig i
m m
end case
n
Case g i and i S
m
M S M S
m m
W S pullo W S i
m m
end case
Case g ij and i S or j S
m
M S M S
m m
W S W S
m m
end case
Case g ij and i S and j S
m
Comment d i j denotes the distance b etween vertex i and j
if P C W C suchthat i j P C and d i j for G n C
m m m m
M S M S
m m
W S addEdgeOk W S i j
m m
else
M S M S
m m
W S dropBoundary W S i j otherPlus W S i j
m m m end
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
end case
Function pullo W S i
m
new W
for each P C W S
m m
for every pair of b oundary edges i j and i k of n C
P C P C f j k g
m m
end
new W new W P C nf i j j i j g
m
end
return new W
end function
Function addEdgeOk W S i j
m
new W
for each P C W S
m m
if i j P C and d i j for G n C
m m
new W new W P C
m
end
end
return new W
end function
Function dropBoundary W S i j
m
new W
for each P C W S
m m
new W new W P C nfig P C nfj g
m m
end
return new W
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
end function
Function otherPlus W S i j
m
new W
for each k n S
if M S fk g M S
m m
for each P C S S fk g with i j P C
m m m
new W new W P C
m
end
end
end
return new W
end function
return M
n
end algorithm
Theorem Algorithm correctly computes the size of the minimum sized
K cover of a t parse G in linear time
n
Pro of The initial state of G is correct by de nition Wenow show that the state
t
of the pre x G is correctly computed from the current op erator g and the
m m
state of G
m
n
Case vertex op erator i i S
By de nition of the M S s M S nfig M S Let i denote the new vertex in
m m m
V G n V G We rst show M S M S nfig Let C S nfig be a
m m m m
We immediately see that C C fi g is a witness minimum witness cover for G
m
cover for G with M S nfig vertices Now supp ose M S M S nfig
m m m m
and let C b e a witness cover for M S Since C must contain i we see that
m
C C nfi g is a witness cover for M S nfig withfewer than M S nfig vertices
m m
This contradiction shows that M S min M S M S nfig
m m m
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
The pulloff function simply transforms a previous witness cover C represen
tation for G to one for G bymoving the old vertex i into the interior Two
m m
things are done during this transformation any paths of length twobetween
three b oundary vertices with vertex i as the mid p ointisnow added to P C and
m
any ordered pair with vertex i in P C is removed
m
n
Case vertex op erator i i S
Since M S M S fig and W S fig W S ifthe M equality holds w e
m m m m m
just need to consider the previous state restricted to S Clearly M S M S
m m
n
since the op erator i do es not add any edges and the new b oundary vertex do es not
have to b e part of a witness cover C
Case edge op erator ij i S or j S
Since the new edge created by ij is covered byany witness cover C S for M S
m
weseethat M S M S Also since G is a subgraph of G we also have
m m m m
M S M S
m m
ij i S and j S Case edge op erator
Clearly M S M S and M S M S since any witness cover
m m m m
for G containing S is a witness cover for G and at most one more vertex is
m m
ij creates needed to removeanynewK s that the edge op erator
If M S M S then there is a witness cover C S of G suchthatG n C
m m m m
do es not have a path of length twobetween b oundary vertices i and j This can
b e detected by lo oking for a P C such that i j P C and that the remaining
m m
boundary n C do es not have a path of length twobetween i and j Note that if
either i or j is in C then suchaP C satis es these conditions
m
If no such witness cover C exists for G that can b e used for G then all
m m
signi cant witness covers with one more v ertex can b e found bytwoways
including either vertex i or vertex j in the various C covers for M S see the
m
dropBoundary function or lo oking for another b oundary vertex b esides i and
j that can b e safely deleted see the otherPlus function Note that we are not
concerned with deleting a single internal vertex since they do not help in covering
future K s created by the su x g g that is not provided by one of the
m n
ab ove constructed covers
CHAPTER SOME GENERALIZED VC AND FVS GRAPH FAMILIES
The Path and Cycle Cover Obstructions
Figures present the known connected obstructions for the four graph families
Path Cover Path Cover Path Cover and Path Cover
Weknow of at least connected obstructions for the Path Cover family and
at least connected obstructions for the Path Cover family The observant
reader may notice that Path Cover and Path Cover share a common
obstruction
Figure presents the seven connected obstructions for the Cycle Cover
graph family There are at least connected obstructions for the Cycle Cover
graph family Oddly enough the Cycle Cover and P ath Cover fami
lies havetwoidentical obstructions
Figure Connected obstructions for Path Cover
Figure Connected obstructions for Path Cover
Figure Known connected obstructions for Path Cover
Figure Known connected obstructions for Path Cover
Figure Connected obstructions for Cycle Cover pathwidth
Chapter
Outer Planar Graphs
Since several surface emb edding families of graphs are closed under the minor or
der the main purp ose of this chapter is to develop our b ounded combinatorial width
technique for characterizing these typ es of lower ideals This chapter uses an approx
imation search via universal distinguishers to nd a set of forbidden minors for a
simple generalization of the outer planar graph family
Intro duction
De nition A graph G is outer planar if there exists a planar drawing of G
with one face region containing all the vertices of G The family of graphs that are
within k vertices of outer planar is de ned as follows
k Outer Planar
fG j there exists a V V G such that G n V is outer planar and jV j k g
One reason that the class of outer planar graphs is p opular is that the art of
drawing them is in uenced bymany computable criteria most of which are based on
geometric symmetries Man It would b e interesting to know whether the same
metho ds apply to the k Outer Planar families
It is known that there are two obstructions to the family of outer planar graphs
namely K and K see for example Tru The computational complexit y for
CHAPTER OUTER PLANAR GRAPHS
outer planar memb ership problem is easily seen to b e linear time by using one of the
known linear time planarity algorithms along with the next result Other linear time
outer planar algorithms are presented in Mit and SI
Observation A graph G is outer planar if and only if G K is planar
Pro of Recall that G K denotes the graph created by adding a new vertex to G
of degree jGj IfG is outer planar then there exists a planar emb edding of G K
by placing K in the outer face of an outer planar emb edding of G If G K is
planar then let S v v v beneighb ors of K in clo ckwise order in a planar
jGj
emb edding E of G K The face F where S is a subsequence that contains K in
E n K is an outer face of an outer planar emb edding of G
Another useful result states that ev ery biconnected outer planar graph with at
least vertices has a unique Hamiltonian cycle CB
The Outer Planar Computation
Wenow turn our attention to the family of graphs Outer Planar i e those
graphs that are within one vertex of outer planar Using the ab ovelemmawehave
a practical O n memb ership algorithm Wenow state some direct nonminimal in
the minor order pretests
Lemma If a connected graph satis es any of the following prop erties then it
is not an obstruction to Outer Planar
Contains adjacent degree twovertices
Contains a cut edge e g contains a degree one vertex
Contains a cut vertex i e not biconnected
Pro of Assume u and v are adjacent degree twovertices of an obstructions G Let
be the u v edge contracted minor of G with the new vertex lab eled w Since G
CHAPTER OUTER PLANAR GRAPHS
G Outer Planar there exists a vertex x of G such that G G nfxg is
outer planar If x w then G nfug or G nfv g would b e outer planar Also x is not
adjacentto w since removing x from G would also showthatG Outer Planar
the p endent edge incidenttovertex w can b e sub divided Likewise if x is not
adjacentto w which means x is not adjacenttoeitheru and v then the graph
G nfxg can b e emb edded in the plane such that the degree twovertex w is on the
outer face Expanding w to its original u v edgegives a contradiction to the fact
that G Outer Planar
Now consider u v to b e a cut edge of an obstruction G Let C and C be
u v
the two comp onents of G nf u v g Since G is an obstruction exactly one of these
comp onents must b e in Outer Planar and not outer planar while the other is
outer planar Without loss of generality assume C is outer planar If C nfv g is
u v
outer planar so would G nfv g Thus an interior vertex w w v ofC must b e
v
C s witness for b eing in Outer Planar However the graph C nfw g has an
v v
emb edding with v on the outer face and an edge added b etween u v shows that
G nfw g is outer planar Therefore any obstruction to Outer Planar can not
have cut edges
At last consider a vertex v to b e at least a three waycut vertex of an obstruction
G Let C C C m b e the comp onents of G nfv g andC C C
m
m
be the vertex induced comp onents of G with vertices fv g V C i m We
i
assume that every comp onent C has at least one edge for otherwise wewould have
i
a cut edge Since G is not within one vertex of outer planar at least one of the
is outer planar is not outer planar If anygraph C C and hence one of the C
i
i i
let G b e the Outer Planar graph created by deleting an edge e of C from G
i
Notice that if G nfv g is outer planar then so is G nfv g Let w w v beavertex
of G such that G nfw g is outer planar Since the induced subgraph C is already
i
outer planar w must b e in one of the C j i Since we can put back the edge
j
e in G andhave an outer planar emb edding of G nfw g this implies that eachof
the C is not outer planar Let G Outer Planar b e the graph G with an
i
v is the only edge e of C deleted Since v is the only shared vertex b etween the C
i
C C p ossible witness to G b eing in Outer Planar Thus C must all b e
m
in Outer Planar Ifwe rep eat the ab ove argumentfor e C then C is also
CHAPTER OUTER PLANAR GRAPHS
shown to b e in Outer Planar In these sub cases vertex v is the witness and
thus v would b e a witness for G Outer Planar However since G is assumed
to b e an obstruction wehave a contradiction to the fact that C exists Therefore
G can not have three way cut vertices
Wenow assume an obstruction G has a two waycut vertex v Again consider C
and C the comp onents of G nfv g and C and C as describ ed ab ove Let u v be
an edge in C The graph G G nf u v g is in Outer Planar There exists a
witness vertex w of G suc h that G nfw g is outer planar Vertex w must b e in C for
would b e outer planar And w v for otherwise G would b e a member otherwise C
v in of Outer Planar Sow C Ifwe rep eat this pro cess with an edge u
C we get another witness vertex w in C for G nf u v g Since C C nf u v g
and C C nf u v g we see that G nfv g must b e outer planar This contradicts
the fact that G is an obstruction So G can not haveany cut vertices
The following result given with an alternate pro of is well known for example
it is stated in Van Leeuwin s handb o ok without a pro of vL
Lemma Any outer planar graph has treewidth at most
Pro of Clearly the complete graphs K and K have treewidth at most Without
loss of generality assume that wehave an outer planar graph G that is triangulated
since removing edges from a graph only decreases its treewidth The graph G has
a degree twovertex v since it would otherwise contain the complete graph K as a
minor see Ram w e knowthat K is an obstruction to outer planar Consider
the outer planar subgraph G G nfv g By induction G has a tree decomp osition
of vertex sets fV g i I of width Let e x y b e the internal non b oundary
i
edge of vertex v s triangle in G SomeV in the tree decomp osition for G contains
i
y We can create a tree decomp osition of G of width by adding both vertices x and
aleafvertex set V fx y v g and attaching it to V
v i
Corollary Any obstruction of Outer Planar has treewidth at most
Pro of Here any outer planar graph has treewidth at most For such a tree decom
p osition we can include two additional vertices in each set to accountforany xed
CHAPTER OUTER PLANAR GRAPHS
vertex in a deleted vertex minor and one vertex for the witness within one vertex
With the use of the ab ovetwo results we can givea weak pathwidth b ound for
Outer Planar s obstruction set
Lemma If G is an obstruction to Outer Planar and has pathwidth greater
than then G has at least vertices
Pro of Let G G nfv g b e a minor of G for anyvertex v G Observe G
is connected by Lemma Since G Outer Planar there exists a vertex
w G such that the graph G G nfw g is outer planar If G has pathwidth
then we can combine vertices v and w with all the vertex sets in a path decomp osition
of width for G to get a path decomp osition of width for G
Now consider the following t wo p ossibilities for G to havepathwidth at least
If G contains a biconnected subgraph S of pathwidth then consider the dual
graph D of S without the outer face vertex using any induced outer planar emb edding
of G Without loss of generality we can assume that S is triangulated It is easy
to see that the graph D must b e a tree and have maximum degree three The
corresp onding pathwidths of the S s for the p ossible trees D see Figure of
order are all less than Thus since jD j jS j we know that the smallest S
with pathwidth has vertices see Figure So the obstruction G then has to
have at least vertices
The other p ossibility is that G must have a cut vertex u and that at least three
of the outer planar comp onents C C C of G nfug must have pathwidth
m
ehave taken care of the case that one of the C has pathwidth or greater W
i
by the argumentabove The smallest outer planar graph of pathwidth is the
vertex augmented graph A K this is the second smallest path width obstruction
the smallest K is not outer planar Thus in this case G must have at least
vertices to have pathwidth greater than
CHAPTER OUTER PLANAR GRAPHS
Figure All vertex outer planar dual trees duals minus outer face vertex
Figure The smallest outer planar graph with pathwidth
CHAPTER OUTER PLANAR GRAPHS
An outer planar congruence
Observe that a disconnected graph is outer planar if and only if each comp onentis
indep endently outer planar Thus without loss of generality assume for each pre x
G of a t parse G at most one comp onent has order greater than one
m n
Wenow de ne an outer planarity state function f from t parses to states Our
outer planarity congruence on t parses G H is based on equality of states f G
o
f H
An outer planarity state consists of a set of outer planar emb eddings and each
outer planarity embedding consists of these ingredients
Clo ckwise circular sequence Seq of b oundary vertices of the outer face with
duplicates allowed but minimize in length as indicated b elow
A b oundary indicator vector Gap Are there hidden interior vertices lo cated
between the gaps of the b oundary sequence Seq Here jGapj jSeqj The
elements of a gap vector contain any of the following symb ols
a E just an edge
b P path of non b oundary vertices
c C cycle face b etween vertices
An optional add edge list Add Which b oundary edges can b e added and still
t
preserve the outer planarityemb edding Note jAddj
Tosave on storage one maywanttokeep only canonical Seq s and do pancake
ips of the circular sequence of the emb eddings kept This makes the congruence
a bit more complicated and so we do not discuss this optimization further Let l G
t
b e the maximum length of Seq over all outer planarityemb eddings of a t parse G
The following lemma shows that the range of this function is nite
t if t
Lemma For any t parse G l G
t
t if t
CHAPTER OUTER PLANAR GRAPHS
E
E
v
v
v
P
s
i
s
i
b oundary vertex
u
internal vertex
v
s
j
s
j
Figure Illustrating the pro of of Lemma
Pro of Let b t denote the number of boundary vertices For b the
comp onentmust b e K so it trivially follows that l G Likewise for b the
t parse is acyclic so l G
For a xed outer planarityemb edding supp ose that the same b oundary vertex v
app ears on the outer face twice without another b oundary vertex b etween them The
internal vertices b etween these consecutive lo cations in Seq form a lo op with p ossible
cords see upp er part of Figure This means that these internal vertices can b e
emb edded merged into any extended outer planar emb edding of the graph minus
the lo op Thus consecutive identical b oundary vertices in Seq are not needed If
b then accounting for b oundary rep etitions e g vertex u in the lower part of
Figure wehave the following b ound
l G max fl G m g
t t m
m
The worst case is when m So after solving the recurrence relation for l G
t
l G we can conclude that l G t for t
t t
Observe that equality holds in the ab ovelemmawhen G is a star Furthermore
we conclude with the following corollary
CHAPTER OUTER PLANAR GRAPHS
Corollary There are only a nite numb er of outer planarityemb eddings for
the family of t parses And hence only a nite numb er of p ossible outer planarity
states with resp ect to
o
Pro of The number of combinations of the Seq and Gap vectors is b ounded ab ove
P
t
i i
by i The add edge list Add is uniquely determined from Seq and Gap and
i
do es not add to the numb er of states
A nite state algorithm
Wenow present a nite state algorithm that recognizes those connected pathwidth
t parses that are outer planar This dynamic programming algorithm nds the outer
planarity state of a t parse G by nding in order the outer planarity states of
n
the pre x t parse s G G G where the subscript n denotes the number of
n
op erators scanned Recall the outer planar state function f from Section for
At parse G is outer planar if and only if f G is non empty
o
As mentioned in the previous section we assume that only one comp onentof
any pre x t parse has order greater than one Any connected t parse can b e easily
converted to this format
n n
For the base case consisting of an empty t parse G t the initial
state F f G is the set of singleton outer planarityemb eddings f Seq i Gap
E j i tg The gap E acts as a lo op which is dropp ed later
For the inductive case wenowshowhow to nd the outer planarity state F
m
f G from the outer planarity state F f G If g is a vertex op erator at
m m m m
p osition m of G wedoCase below
n
n
Case g i
m
F
m
For all emb eddings E F do
m
n
For all vertices v i Seq E
j
Gap E M er g eGaps Lef t Gap E j v Right Gap E j
j
end
Seq E Seq E nfig
CHAPTER OUTER PLANAR GRAPHS
Table The MergeGaps function for outer planar Case
Lef t Gap Rig ht Gap Resulting Gap
E E P
E P P
E C C
P E P
P P P
P C C
C E C
C P C
C C C
Add E f j k j j k Add E andi j and i k g
F F E
m m
end
end case
Note that if vertex j is lo cated b etween twovertex i s in Seq E then two j s are
not adjacentinSeq E Now consider an edge op erator at p osition m
ij Case g
m
F
m
For all emb eddings E F do
m
if vertices i and j are in the same comp onent
if i j Add E
Seq E Seq E
if gap consists of fE jP g
Gap E Path C y cl e Gap E i j
else
Gap E Gap E
end
Add E Add E n C r ossing s Seq E i j
CHAPTER OUTER PLANAR GRAPHS
F F E
m m
end
else if b oth i and j are isolated i e the rst edge
Seq E i j
Gap E E E
Add E
F F E
m m
else attaching isolated vertex say j to comp onent
n
For all vertices v i Seq E
k
Seq E Lef t Seq E k j Rig ht Seq E k jSeq E j
Gap E I nser tE dg eGaps Gap E k
Add E Add E f j v j v is fE jP g away from ig
F F E
m m
end
end
end
end case
Note that weemb ed p endentvertices to the outside face The gap vector is
used in the C r ossing s routine to determine if future edges may b e added b etween
b oundary vertices without hiding internal vertices that are currently on the outer
face
Wenow illustrate with the ab ove outer planarity algorithm as our example a
standard pro of technique for proving correctness of dynamic programming algorithms
on t parses
Theorem Our nite state outer planarity algorithm is correct
Pro of sk etch For any t parse G with a set of outer planar emb eddings E let
m m
f G E IfG is not outer planar then the set E is empty For any xed em
m m m m
b edding E E of G let g E R where R is a reduced outer planar emb edding
m m
describ ed ab ove represented by our nite state algorithm For a set of emb eddings
E we also use the notation g E to denote the image R of the map E onto
m m m m
CHAPTER OUTER PLANAR GRAPHS
fg E j E E g For any t parse op erator g in we de ne three meta functions
m m t
A A and A that up date t parses outer planar emb eddings and reduced outer
planar emb eddings resp ectively with the obvious actions for the di erent g See
m
Figure The function A is our usual t parse building function
t
t t
The function A takes eachemb edding E of E and an op erator g and returns all
m m
p ossible extended emb eddings If g is a vertex op erator then A E g returns
m m
exactly one emb edding while if g is a edge op erator then A E g may return
m m
zero or more emb eddings The function A is the nite state algorithm that we are
proving correct
G
G
m
m
A
f f
E
E
m
m
A
g g
R
R
m
m
A
Figure A commutative diagram for the pro of of Theorem
Notice that if a t parse G has an outer planar emb edding the set E is non
n n
empty and furthermore by the de nition of the function g the set R is non empty
n
Otherwise if G is not outer planar then b oth E and R are empty Thus in order
n n n
to show that the outer planar t parse algorithm A is correct all that we need to do
is show that the diagram in Figure commutes for each t parse op erator in We
t
leave this messy case analysis to the reader
CHAPTER OUTER PLANAR GRAPHS
The Outer Planar Obstructions
Using a universal distinguisher search pathwidth our list of obstructions for
Outer Planar stabilized to the ones listed in Figures and With con
secutive randomized runs o ccurring without new obstructions app earing we b elieve
that this set is close to b eing complete Each run takes ab out three weeks to complete
using roughly distributed worker pro cesses to prove minimality or guess nonmini
malityofthe t parse s
Figure Known connected obstructions for Outer Planar
Figure Continued Outer Planar obstructions
Some other surface obstructions
Besides the previously known sets of planar outer planar and pro jective
plane minor order obstructions and now our initial set of Outer Planar
obstructions wehave started to nd the obstructions for the within vertex of
planar graph family denoted Planar All of the connected forbidden minors
with at most vertices is displayed in Figures and on the following pages
Besides these obstructions one disconnected obstruction namely K K has
vertices The other two disconnected obstructions K K and K K have
and vertices resp ectively We exp ect that the complete set of obstructions will
contain ab out graphs Thus based on our feasibility conjecture this maybethe
only within k vertices of planar family that can b e computed during the foreseeable
future It is interesting to see that of the linkless emb edding obstructions recall
Figure are also obstructions for this almost planar graph family
For the next grand challenge surface to characterize Neufeld has computed a
partial list of toroidal obstructions i e minor order forbidden torus emb eddable
graphs based on a brute force search Neu Within his set of just over two thousand
obstructions the densest one known has pathwidth See Figure for the drawing
of this symmetric obstruction which surprisingly is the complement of the famous
Petersen graph There are exactly and minor order obstructions with
and vertices resp ectiv ely Neu The eightvertex obstructions are shown in
Figure Since it app ears to b e impractical to search through all the graphs on
or more vertices our computational metho d may b e the only feasible approachto
classify this surface However there is a big gap b etween our pathwidth lower b ound
and the currently b est known combinatorial width upp er b ound e g treewidth
given by Seymour Sey
To tackle the torus wehave a nite index congruence for any xedgenus k based
on Eulor s surface formula n m f k where f is the numb er of faces
of a genus k emb edding but unfortunately it is not practical for the anticipated
pathwidth b ound Also wehave b een workingona t b oundaried graph testset but
it also seems to b e impractical i e to o large at the moment
Figure All connected obstructions with at most vertices for Planar
Figure Continued small obstructions for Planar
3 1
6 4
2 4
8
9 5
0 5
6 8 7
9 3
7
0 1 2
Figure A dense symmetric toroidal obstruction with pathwidth
K n E C K n E K K n E K E P
Figure The three smallest vertices toroidal obstructions
Part III
Applied Connections
Chapter
Pathwidth and Biology
In this short chapter weshow that an imp ortant problem in computational biology is
equivalent to a colored version of a well known graph layout problem In order to map
the human genome biologists use graph theory particularly interval graphs to mo del
the overlaps of DNA clones cut up segments of a genome Mir For engineers
VLSI circuits must b e laid out in order to minimize physical and cost constraints
The vertex separation see b elow of a graph layout is one such measurementofhow
go o d a layout is i e the di erence b etween the numb er of tracks required for the
layout and the graph s pathwidth
VLSI Layouts and DNA Physical Mappings
The NP complete combinatorial problem of Intervalizing a ColoredGraph ICG
rst de ned in FHW and indep endently given in GKS as the Graph Interval
Sandwich problem is intended to b e a limited rst step mo del for nding DNA
physical mappings For this mo del it is assumed that the biologist knows some of
the overlaps for instance overlaps sp eci ed by some probability threshold based on
the physical data The question asked by the ICG problem is whether other edges
can b e prop erly added to di erently colored vertices to form a colored interval graph
The V ertex Separation VS graph problem is related to manydiverse problems
in computer science b esides its imp ortance to VLSI layouts Lengauer showed that
CHAPTER PATHWIDTH AND BIOLOGY
solving the progressiveblack white p ebble game imp ortant to compiler theory and
determining the vertex separation for a graph are p olynomial time reducible to each
other Len No de searchnumb er a variant of searchnumber Par was shown
equivalenttothevertex separation plus one by Kirousis and Papadimitriou KP
From EST the searchnumb er is informally de ned in terms of p ebbling to b e
the minimum numb er of searchers needed to capture a fugitive who is allowed to
move with arbitrary sp eed ab out the edges of the graph For no de searchnumb er
a searcher blo cks all neighb oring no des without the need to move along an incident
edge The exact details of the ab ovetwo p ebbling games may b e found in Bie
Kinnersley in Kin has shown that the pathwidth of a graph is identical to the
vertex separation of a graph The concept of pathwidth has b een p opularized bythe
theories of Rob ertson and Seymour see for example RS b Thus since the gate
matrix layout cost another well studied VLSI layout problem KL M oh equals
the pathwidth plus one FL b it also equals the vertex separation plus one
Our result shows that the Vertex Separation problem is imp ortant to another
area b esides computer science namely computational biology
An Equivalence Pro of
In this section we formally de ne two parameterized problems k ICG and k CVS and
then show that they are indeed equivalent
De nition A layout L of a graph G V E is a one to one mapping L
V f jV jg
If the order of a graph G V E isn weconveniently write a layout L as a
permutation of the vertices v v v For anylayout L v v v of G
n n
let V fv j j i and v v E for some k ig for each i n
i j j k
ation of a graph G with resp ect to a layout L is De nition The vertex separ
vs L G max fjV jg Thevertex separation of a graph G denoted by vs G
i jGj i
is the minimum vs L G over all layouts L of G
CHAPTER PATHWIDTH AND BIOLOGY
The k coloring of a graph G V E is a mapping col or V f kg For
any subset V V let Colors V fcol or v j v V g
De nition A colored layout L of a k colored graph G V E islayout L
such that for all i n col or v Colors V
i i
Problem Colored Vertex Separation CVS
Input A k colored graph G
Parameter k
Question Is there a colored layout L of G where vs L G k
Problem Intervalizing a Colored Graph ICG
Input A k colored graph G V E
Parameter k
Question Is there a prop erly colored sup ergraph G V E ofG E E such
that V V and G is an interval graph
In Figure weshow a colored graph with an interval sup ergraph represented
on the left and a colored vertex separation layout given on the right The dashed
edge b etween the di erent colored vertices is not an edge in the input graph
Theorem For any xed p ositiveinteger parameter k b oth k CVS and k ICG
are identical problems
Pro of Let L v v v b e a colored layout of a k colored graph G V E
n
We showhow to construct a prop erly colored sup ergraph G that is also an interval
hvertex v V de ne the interval graph For eac
i
I a b i maxfj j v v E or j ig
v v v i j
i i i
By de nition if edge u v E then I I Now de ne the sup ergraph
u v
G V E where an edge v v E whenever I I It su ces to show
i j v v
i j
v inE n E Without loss of generality that col or v col or v for eachedge v
i j i j
assume i j so that b a Again by the de nition of I there exists a vertex
v v v
i j i
CHAPTER PATHWIDTH AND BIOLOGY
White
Black
Gray
a Interval sup ergraph
b Linear CVS layout
Figure Illustrating the k CVS and k ICG problems
v such that j kand v v E This implies that v V This also holds
k i k i j
for i j Now L is a colored layout so col or v Colors V Thus
j j
G that is col or v col or v Therefore G is a prop erly colored sup ergraph of
i j
intervalizable
For any k colored graph G V E that satis es ICG let fI j v V g be
v
an interval graph representation of a sup ergraph G V E Let a b be the
v v
endp oints of the interval I a b forvertex v Without loss of generality assume
v v v
that a a implies u v Let L v v v b e the unique layout such that
u v n
To prove i j if and only if a a We claim that L is a colored layout of G
v v
i j
this claim weshow that col or v Colors V i n If there exists a vertex
i i
u V suchthat col or u col or v thenby de nition of V vertex u must b e
i i i
CHAPTER PATHWIDTH AND BIOLOGY
adjacenttoavertex v for some j i Further j i since the edge u v
j i
would not b e a prop erly colored edge Since a a and u v E wemust have
u v j
j
b a in order to form an overlap However b a a These inequalities
u v u v v
j i j
hold since the intervals u and v with the same color do not overlap and i j
i
Wehave reached a contradiction So u V if col or u col or v Thus L is a
i i
colored lay out
Now supp ose that for some r sthere exist twovertices v and v in V with
r s i
the same color Since v V there exists a vertex v with j isuch that v v
r i j r j
E This implies v V But this implication contradicts the fact col or v
r s s
C olors V So col or v col or v Hence any set V fv g has at most one
s r s i i
vertex of each color Since there are k colors each V must have k orfewer vertices
i
Thus vs L G vs L G k
Some Comments
Recently the corresp onding general problem of intervalizing a colored graph to a unit
prop er interval graph has b een shown to b e NP hard and xed parameter hard for
W by Kaplan and Shamir KS also see GGKS KST The go o d news
from Kaplan and Shamir s pap er is that for each parameter k i e k colors this unit
interval problem has a p olynomial time decision algorithm
More recently determining whether a p olynomial time decision algorithm exists
for k ICG or equivalently k CVS has b een solved by Bo dlaender and de Fluiter
BdF Their main results are a quadratic time algorithm to test if a colored
graph is a subgraph of a prop erly colored interval graph and an NP completeness pro of for this problem for colors
Chapter
Automata and Testsets
The main result of this chapter shows that nding a minimum sized testset for any
minimal nite state automaton is NP hard Also given is a simple graph algorithmic
example whichtakes a b oundaried graph testset and builds an automaton Atthe
end of this chapter weshowhow these b ounded width memb ership automata for
lower ideals can b e e ciently used to nd obstruction sets
Intro duction
The reader mayhave seen the following well known result from classical automata
theory e g see HU
Theorem Myhill Nero de The following two statements are equivalent
A set L is accepted by some nite state automaton That is the set L is
a regular language
Let b e an equivalence relation for x and y in de ned by x y if and
L L
only if for all z in xz is in L exactly when yz is in L The relation is of
L
nite index
For any regular language L over the alphab et tw o strings x and y are distin
if xz L and yz L or yz L and xz L Otherwise guished by a string z in
CHAPTER AUTOMATA AND TESTSETS
x and y are said to b e congruent with resp ect to the canonical congruence Aset
L
T is a testset for the language L if every two strings x and y of that are in
di erentequivalence classes of are distinguished by some test z in T
L
The pro of of the Myhill Nero de Theorem shows that if M is a minimal nite
state automaton for a regular language L then there is a one to one corresp ondence
between the states of M and the equivalence classes of
L
Since only one test is needed to distinguish anytwo states of an automaton M
jM j
clearly a testset T need not contain more than tests Likewise since a single
test can only partition the states into two classes a testset T needs to contain at
least dlog jM j e tests An obvious problem arises on how to nd the smallest such
testset for an arbitrary regular language L Regrettably we present some negative
results concerning this task in Section
To illustrate the relationship b etween testsets and regular language decision prob
lems we give in Section a testset for the graph connectivity decision problem for
graphs of b ounded pathwidth and treewidth In fact many graph decision problems
for b ounded pathwidth or treewidth can b e solved in linear time by the use of au
tomata e g see AF There are no large hidden constants in the running times
of these algorithms A graph building alphab et e g the alphab et for t parses is
t
used to represent the graphs of b ounded combinatorial width Each graph contains a
sp ecial set of b oundary vertices where tests can b e app ended In this context these
b ounded width graph families are regular languages There are many applications for
these nite state automata such as testing for VLSI layout prop erties e g see
KK b and nding obstruction set characterizations which is brie y discussed
in Section
Finding a Minimum Testset is NP hard
This section shows that for an arbitrary minimal nite state automaton determining
the size of a minimum testset is at least as hard as any problem in the complexity
class NP A formal de nition of our testset problem now follows
CHAPTER AUTOMATA AND TESTSETS
Problem Automata Minimum Testset AMT
Input A minimal deterministic nite automaton DFA M Q Q
Q q Q F Q and a p ositiveinteger k
Question Do es there exist a testset T for M with jT j k That is for a
solution T and all q q Q there exists a test t T suchthat q t F and q t
i j i j
F or q t F and q t F where q denotes any equivalence class representative
i j
for the state q
If we can solve this AMT problem in p olynomial time for each input k then we
can determine in p olynomial time the least k such that an automaton M has a testset
of cardinality k i e the optimization problem would also b e in the complexity class
P A related and classic problem is the following
Problem Minimum Test Collection for sets MTC
Input A nite set S of p ossible diagnoses a collection C of subsets of S repre
senting binary tests and a p ositiveinteger j jC j
Question Do es there exist a subset C of C jC j j such that for every pair x y S
there is some test c in C for which jfx y g cj here a test c eliminates one of
the two diagnoses x and y
Our main result of this chapter is now presented
Theorem The decision problem AMT is NP hard
Pro of Gary and Johnson haveshown that MTC is NP complete GJ Weshow
that AMT can b e restricted to MTC This then shows that AMT is NP hard
Given an instance S C of MTC with j jC j we build an automaton M
For convenience we assume relab el if Q q F as displayed in Figure
necessary that the nite set S equals f ng The state q Q corresp onds to
i
a particular i S for i n
We rst show that the automaton M is minimal Let L b e the language accepted
by M Our automaton is nonminimal if and only if there exists two states q and q
such that q z L q z L for all z The empty test distinguishes F from
CHAPTER AUTOMATA AND TESTSETS
Q fq j i ng fq g fq g
i f d
f ng ft j c C g f g
c
S
F Q nfq q g C ft j i cg f g
d i c
c C
Figure The AMT automaton M built from an MTC instance
fq q g The test distinguishes state q from the states q q q and b etween
d f n
states q and q Thus the only way that the automaton M is nonminimal is if
d
there are two diagnoses in S that share identical tests in C In this case whichis
easily checked in p olynomial time there do es not exist any C C to solve the MTC
problem Hence hereafter M is assumed to b e minimal
To complete the reduction we set k the inputed p ositiveinteger for AMT to b e
j where j was the queried p ositiveinteger for MTC We claim the MTC problem
has a collection C C suchthat jC j j if and only if the AMT problem has a
testset T such that jT j k
suchthat jC j j We Assume that the MTC instance has a subset C of C
claim that the set T ft j c C g f g is a testset for M The tests and
c
are su cient to distinguish states q q and q from the rest of Q
d f
Test distinguishes q from Q nfq g and test distinguishes q and q
d f d f
CHAPTER AUTOMATA AND TESTSETS
Test distinguishes q from Q nfq g and test distinguishes q and q
f f
Test distinguishes q from Q nfq g and test distinguishes q and q
f f
States q and q a b n are distinguished by some t since jfa bg cj for
a b c
S
some c in C Iffa bg c fag then by the de nition of C ft j i cg f g
i c
c C
wehave q t L and q t L SincejT j k the corresp onding AMT instance is
i c j c
also true
Wenow show that if the AMT instance is true then the MTC instance must also
b e true First note that any test pre xed with a symbol t do es not help in distinguish
c
ing the three states fq q q g Similarly any test pre xed with f jS jg do es
f d
not help in distinguishing fq j i S g Since all of the p ossible tests in havebeen
i
exhausted weknow that it requires indep endent tests T T T to distinguish all
of the states of Q that is a set T for fq q q g and a set T for fq j i S g
f d i
At least two tests are needed to show that the three states q q and q are
f d
not equivalent since dlog e So without loss of generality we can assume
T f g distinguishes states q q andq from the rest of Q This implies that
d f
j or fewer tests is su cient for a complete T Any signi canttestinT must b e
of length for otherwise it do es not distinguish anything in fq j i S g Those
i
j j tests in T that distinguish b etween all the states of S corresp ond to j subset
collections of C Thus the MTC instance must b e true
ve construction builds a nite automaton that has a minimum testset The ab o
with single character tests It would b e interesting to nd a construction that b ounds
the alphab et size j j while allowing longer test lengths since more natural problems
seem to b e of this typ e
It is still an op en problem whether AMT is NP complete Two standard means
of attack for the general problem are to show AMT is in NP and to nd a
reduction from AMT to a known NP complete problem For metho d wehave
the following result
Lemma Given an automaton M Q q F there exists a test of length
at most jQj jQj to distinguish anytwo states of M
CHAPTER AUTOMATA AND TESTSETS
Pro of For anytwo non equivalent states q and q let t a a a a a be a
i j n
distinguisher where each a For any pre x t a a a of t the state q t
i i i i
must b e di erent than the state q t since the automaton M is deterministic and
i
this would force q t q t If the state pair q t q t equals q t q t for some
i i j j
j i j n thent a a a a a would b e a shorter test Thus the
i j n
numb er of distinct ordered state pairs jQj jQj is an upp er b ound on the needed
test length for two non equivalent states
If we are interested in a restricted version of the AMT problem where the tests
have a b ounded p olynomial length then this problem is NP complete In lightof
the ab ove fact we de ne the following problem
Problem Bounded Automata Minimum Test Set BAMT
Input A minimal deterministic nite automaton DFA M Q q F and a
p ositiveinteger k
Question For the automaton M do es there exist a testset T with jtj jQj
jQj for all t T and jT j k
Corollary The decision problem BAMT is NP complete
Pro of First observe that the length of any useful test t T given in the pro of
of Theorem has a b ounded length of So using the previous restriction to
MTC instances we see that BAMT is NP hard All that remains is to show that in
p olynomial time one can verify a solution to BAMT For each test t T one needs to
jM j
states of the automaton M to verify that each pair of states has check at most
a distinguisher It takes at most jM j jM j steps to run each test through the
automaton since that is the maximum length of an ytestt T Thus the veri cation
takes p olynomial time
CHAPTER AUTOMATA AND TESTSETS
ATestset Example
Building Memb ership Automata
Wenow present a testset application for input graphs of b ounded combinatorial width
Many di erent sets of graph building op erators can b e used to construct graphs as
wesaw in Section We illustrate the following application with our quadratic
sized op erator alphab et V E for pathwidth t parses see Chapter Later
t t t
in Section wedevelop a setting for the b ounded treewidth case
Assume that wehave a graph family F with a nite index canonical congruence
This implies that the b ounded width canonical congruence over the pathwidth
F
is a regular language Recall op erator set is nite index i e this family F
t
t
from Section that there is a slight distinction b etween these two canonical congru
ences In the b ounded width case a language F is a subset of the graphs t parses of
pathwidth at most t One can view a test Z for as any t b oundaried graph
F
preferably representable by a string of op erators Recall that a t parse G and a test
Z are combined to form a new graph G Z by taking the union of the underlying
b oundaried graphs G and Z except that the b oundary vertices are coalesced Wesay
that G passes the test Z if G Z is in F Note that we could just as easily use the
de nition G passes the test Z if G Z is in F where we use the concatenation
t
op erator for t parses instead of the circle plus op erator
Wenow present a simple example Consider the family F of connected graphs of
pathwidth at most t and order at least t For eachmember G of F assign a vertex
b oundary of size t that corresp onds to a b oundary of a t parse representation of
G We see that the numb er of equivalence classes for equals the number of set
F
partitions of f tg plus one The equivalence class for an arbitrary t
b oundaried graph G is determined as follows
If the graph G has a comp onent without any b oundary vertices then it is in a
dead state equivalence class
partition the set of t b oundary vertices i e Otherwise the comp onents of G
b oundary vertices u and v are contained in the same set of the partition if there
CHAPTER AUTOMATA AND TESTSETS
Figure A graph connectivity testset for b oundaried graphs
exists a path b etween u and v in G Take this partition as the equivalence
class index for G
It is easy to see that if two b oundaried graphs G and H fall into di erent equivalence
classes then there exists a test Z consisting of only edges such that exactly one of the
two graphs G Z and H Z is connected Since these small tests can b e represented
as t parses the b ounded width canonical congruence is equivalentto We use this
F
observation to build a nite testset for the graph connectivity problem for anysetof
k b oundaried graphs
Example The set of ve b oundaried graphs displayed in Figure is a
testset for graphs within A parse is connected if and only if is in the same
equivalence class that is indexed by passing all of the tests in particular passing the
rst test implies passing the other tests
Generally sp eaking if we know a testset for the family F and a memb er
ship algorithm for F then we can compute a minimal nite state automaton that
recognizes any graph of pathwidth t within F This pro cess is mentioned shortly
The next example presents the constructed automaton obtained from the graph con
nectivity testset of Example
Example Consider the alphab et the alphab et for parses A pic
torial display of the automaton generated from the connectivity testset given in
CHAPTER AUTOMATA AND TESTSETS
dead state
n n n
State n Symbol
initial state
dead state
another state
another state
another state
accept state
Figure A graph connectivitymemb ership automaton for parses
Figure is shown ab ove in Figure Note that the missing arcs go to the dead
state Listed b elow the automaton is the transition diagram Q Q
We construct memb ership automata such as the one just presented by the fol
lowing simple rules The two ingredients in building an automaton M for accepting
t parses with resp ect to a nite state graph family F are
A testset T of t b oundaried graphs for the canonical congruence
F
Any memb ership algorithm A for recognizing graphs in F
During the building pro cess of M wekeep an equivalence class representative G
i
t parse for each state q of M as encountered Initially for the start state q the
i
n n
smallest t parse G t is our only representative Next for each op erator
wecheckif G is congruent via the testset to an y state representative
t
kept so far Recall that we can easily check whether two t parse s G and G are
CHAPTER AUTOMATA AND TESTSETS
congruent G G by using T and A
F
for every T T G T F G T F G G
i i i F
If it happ ens that G is congruent to some state representative G then a transi
i
tion entry from q toq is added to the transition function Otherwise G
i
must b e a representative for a new state q We rep eat the pro cess of app ending each
i
single op erator to each new state representative G as they are discov ered
t i
and do congruence checks for G against all previously encountered state repre
i
sentatives The automaton is built when all the p ossible op erator transitions from the
representatives for each G lead to previous states i e when the transition function
i
is not a partial function It is safe to terminate the construction at this p oint since
every state of is reached from the start state by some string of op erators
F
Using tree automata for b ounded treewidth
We can easily extend the previous testset application for treewidth t parse s Here we
build a tree automaton instead of a linear automaton from a testset for the canonical
congruence General information ab out tree automata may b e found in the survey
F
paper Tha We are interested in ro oted tree automata that evaluate their input
arguments parse trees in a leaf to ro ot order For treewidth t parse s wehavethe
following customized de nition
De nition A tree automaton M Q f g q F for the
t
treewidth t parse alphabet is an extended linear pathwidth automaton Q q F
t
where
Q is the set of states
is the alphab et for the input trees
is a function from Q V E to Q i e is a transition function for the
t t
unary op erators
t
is a function from Q Q to Q i e is a transition function for the binary
op erator
CHAPTER AUTOMATA AND TESTSETS
q Q is the start state
F Q is the accepting memb ership states
Just as a linear automaton determines a state for every pre x of an input string
a tree automaton determines a state for every subtree of an input tree of symb ols
while pro cessing the input A t parse G is accepted by a tree automaton M if the
state assigned to the ro ot symb ol op erator is in the set of accepting states F The
rules for evaluating a treewidth t parse G with resp ect to M are de ned recursively
as follows where ev al denotes the evaluation map
M
n n n
If G t then ev al G q
M
If G G G then ev al G ev al G eval G
M M M
If G G g g V E then ev al G ev al G g
n n t t t M M n
An iterativeversion for implementation reasons of this evaluation map ev al is
M
easy to construct Here we rst run the pathwidth automaton starting at eachleaf
op erator until a op erator is reached Then the function is used to merge together
twopathwidth branches or treewidth subtrees The pro cedure then continues up the
input tree assigning states to op erators using b oth the and transition functions
For an example of a tree automata for t parses we extend our graph connectivity
linear automaton of Example
Example A tree automaton that accepts treewidth parses is obtained by
adding the following function to the linear automaton presented in Figure
CHAPTER AUTOMATA AND TESTSETS
Notice that since the binary op erator is commutative the function should
b e symmetric for i e for all x y Q x y y x
F
Before ending this section we need to justify that our de nition of tree automata
for treewidth t parse s is sound The next previously known result shows that it is
su cient to add a transition function to a linear automaton for pathwidth t parses
Lemma For any canonical congruence for a regular language family F if
X X and Y Y then X Y X Y
F F F
Pro of Since X X wehave for each b oundaried graph Z
F
X Z F X Z F
which implies
X Y Z F X Y Z F
By asso ciativityof we see that X Y X Y Likewise since Y Y we
F F
get
Y X Z F Y X Z F
Or Y X Y X Nowsince is b oth a symmetric and a transitive relation
F F
we see that X Y X Y
F
From the ab ove result we can always rst build a memb ership automaton for
pathwidth t parses and then add a transition table for the op erator case to
get a memb ership tree automaton Here the testset metho d of the previous section
is rst applied to pro duce the m jQj equivalence classes of Then for each
F
unordered pair of state representatives G and G i j m we use the testset
i j
to nd the equivalence class of G G Recall that there are at least three avors of
i j
the canonical congruence
F
t b oundaried graph treewidth t parse pathwidth t parse
F F F
We extend a pathwidth t parse automaton to get a treewidth t parse automaton if a
testset for one of the rst two canonical congruences is available
CHAPTER AUTOMATA AND TESTSETS
Quickly Finding Obstructions using Automata
Supp ose wehave a minimal nite state automaton M that accepts t parses in some
minor order lower ideal F It is straightforward as seen b elow to compute the minor
order obstructions of width at most t for F using M This metho d can use either
linear or tree automata
The states of a minimal automaton M for F represent the equivalence classes
of the canonical congruence by the Myhill Nero de Theorem In a breadth
F
rst or depth rst order we start enumerating all t parses p ossibly only the free
b oundary isomorphic t parses as describ ed in Section The search pro cess is
pruned whenever a t parse G has a b oundaried minor that falls in the same state as
G i e a nonminimal t parse is found We can also restrict ourselves to one step
minors as was justi ed in Chapter The minor order obstructions can then b e
extracted from the minimal t parses that fall in the non accept out of family state
of the automaton Since F is a lower ideal there are no transitions out of a non accept
state of M It follows from the de nition of the canonical congruence that there
F
is only one of these out of family states
Notice that a depth rst enumeration order is preferred Wekeep in memory
each t parse G only as long as there is an activ e t parse G that has G as a pre x
parse A t parse in a search tree is called active if it has is not b een proven minimal
or nonminimal
Example Using the nite state automaton for MaxPath over see
Table on page we can compute the single obstruction P a path of length
for the family of graphs that have all paths of length or less A t parse representation
for this obstruction is given b elow
n n n n n n
P
Prefix states
To help followthe t parse P through the MaxPath automaton we listed the
states for eachpre xt parse immediately b elow the last op erator
CHAPTER AUTOMATA AND TESTSETS
Besides these pre x t parse s the smallest search tree contains just two additional
t parses b oth nonminimal namely
n n n n
and
n n n n n
If wehave an automaton for a lower ideal then wemaynotwant to restrict ourselves
to the smallest p ossible search tree since minimality nonminimality pro ofs are easy
to obtain Thus wecanavoid the t parse canonic checks that were mentioned in
Part I of this dissertation Here we do more isomorphism checks at the end of the
obstruction set computation since there are more b oundaried obstructions
The reference counts in Table indicate howmany times each transition was
used during that particular obstruction search Besides every t parse in the search
tree the memb ership automaton is invoked for every one step minor The counts
indicate that many transitions were not used We b elieve that some unnecessary
work was used during the creation of the automaton If the goal is only to compute
obstruction sets for a lower ideal and an automaton is not available we recommend
using the techniques of Chapter
Chapter
Computing Pathwidth by Pebbling
After several rounds of improvement RS b Lag Ree the b est known algorithm
for nding tree decomp ositions is due to Bo dlaender Bo d c For each xed k this
k
n determines that either the treewidth is greater algorithm running in time O
than k or pro duces a tree decomp osition of width at most k By rst running this
algorithm and then applying the algorithm of BK Klo a a similar result holds
for pathwidth Both of the algorithms involved are quite complicated
We describ e in this chapter a very simple algorithm based on p ebbling the graph
k
using a p o ol of O p ebbles that in linear time for xed k either determines that
the pathwidth of a graph is more than k or nds a path decomp osition of width at
most the numb er of p ebbles actually used The main advantages of this algorithm
over previous results are the simplicity of the algorithm and the improvement
of the hidden constant for a determination that the pathwidth is greater than k The
main disadvantage is in the width of the resulting approximate decomp osition when
the width is less than or equal to k
Preliminaries
An homeomorphic embedding of a graph G V E in a graph G V E is
an injection from vertices V to V with the prop erty that the edges E are mapp ed
to disjoint paths of G These disjoint paths in G represent p ossible subdivisions
CHAPTER COMPUTING PATHWIDTH BY PEBBLING
of the edges of G The set of homeomorphic emb eddings b etween graphs is the
top ological partial order that was mentioned in Chapter
Recall that the pathwidth of a graph G is the minimum pathwidth over all path
decomp ositions of G As mentioned in Chapter determining pathwidth is equiva
lent to several VLSI layout problems such as gate matrix layout and vertex separation
M oh EST
Wehaveshown in Chapters and that the family of graphs of pathwidth at
most t t Pathwidth isalower ideal in the minor and hence top ological order
and those graphs with order n have at most nt t t edges
h
Let B denote the complete binary tree of height h and order Let h t be
h
the least value of h suchthatB has pathwidth greater than t andletf t bethe
h t
number of vertices of B To nd a b ound for f t B needs to contain ab ovein
h t h t
the top ological order at least one obstruction of pathwidth t In EST it is shown
that all top ological tree obstructions of pathwidth t can b e recursively generated by
the following rules
The single edge tree K is the only obstruction of pathwidth
If T T and T are any tree obstructions for pathwidth t then the tree T
consisting of a new degree three vertex attached to anyvertex of T T and T
is a tree obstruction for pathwidth t
From this characterization we see that the orders of the tree obstructions of
t
pathwidth t are precisely e g orders and for pathwidth
We can easily emb ed at least one of the tree obstructions for t and
pathwidth t asshown in Figure in the complete binary tree of height t
t t
Thus the pathwidth of the complete binary tree of order f t O is
greater than t
A Simple Linear Time Pathwidth Algorithm
Wesay that a b oundary size k factorization of a graph G is two k b oundaried graphs
A and B suchthat G A B Using the f t b ound given in the previous section
CHAPTER COMPUTING PATHWIDTH BY PEBBLING
Tree Tree
Tree t
Tree t Tree t
Tree t
Figure Emb edding pathwidth tree obstructions Tree t in binary trees
CHAPTER COMPUTING PATHWIDTH BY PEBBLING
the main result of this chapter now follows
Theorem Let H b e an arbitrary undirected graph and let t b e a p ositive
integer One of the following two statements must hold
a The pathwidth of H is at most f t
b The graph H can b e factored H A B where A and B are b oundaried graphs
with b oundary size f t the pathwidth of A is greater than t and less than f t
Pro of We describ e a algorithm that terminates either with a path decomp osition
of H of width at most f t or with a path decomp osition of a suitable factor A
with the last vertex set of the decomp osition consisting of the b oundary vertices
If we nd an homeomorphic emb edding of the guest tree B in the host graph
h t
H then weknow that the pathwidth of H is greater than t During the search for
suc hanemb edding wework with a partial emb edding We refer to the vertices of
B as tokens and call tokens placed or unplaced according to whether or not they
h t
are mapp ed to vertices of H in the current partial emb edding A vertex v of H is
tokened if a token maps to v At most one token can b e placed on a vertex of H at
any given time We recursively lab el the tokens by the following standard rules
The ro ot token of B is lab eled by the empty string
H t
The left child token and rightchild token of a height h parent token P
b b b are lab eled P andP resp ectively
h
Let P i denote the set of vertices of H that are tokened at time step i The
sequence P P P s describ es a path decomp osition either of the entiretyof
H or of a factor A ful lling the conditions of Theorem In the case of outcome
b the b oundary of the factor A is indicated by P s
The placement algorithm is describ ed as follows Initially consider that every
vertex of H is colored blue In the course of the algorithm a vertex of H has its color
changed to red when a token is placed on it and stays red if the token is removed
Only blue vertices can b e tokened and so a vertex can only b e tokened once
CHAPTER COMPUTING PATHWIDTH BY PEBBLING
Algorithm A linear time path decomp osition algorithm
function GrowTokenTree
if ro ot token is not placed on H then
arbitrarily place onabluevertex of H
endif
while thereisavertex u H with token T and blue neighbor v
and token T has an unplaced child T b do
place token T b on v
endwhile
return ftokened vertices of H g
program PathDecomp ositionOrSmallFatFactor
i
P i call GrowTokenTree
until jP i j f t or H hasnobluevertices rep eat
pick a token T with an unplaced child token
remove T from H
if T had one tokened child then
replace all tokens T b S with T S
endif
i i
P i call GrowTokenTree
enduntil
done
end algorithm
Before weprove the correctness of the algorithm we note some prop erties
the ro ot token needs to b e placed step of the GrowTokenTree at most once for each
comp onentof H the Gro wTokenTree function only returns when B has b een
h t
emb edded in H or all parenttokens with unplaced children have no blue neighb ors
in the underlying host H the algorithm terminates since during eac h iteration
CHAPTER COMPUTING PATHWIDTH BY PEBBLING
of step a tokened red vertex b ecomes untokened and this can happ en at most n
times where n denotes the order of the host H
Since tokens are placed only on blue vertices and are removed only from red
vertices it follows that the interp olation prop erty of a path decomp osition is satis ed
Supp ose the algorithm terminates at time s with all of the vertices colored red To
see that the sequence of vertex sets P P s represents a path decomp osition of
H it remains only to verify that for each edge u v of H there is a time i with b oth
vertices u and v in P i Supp ose vertex u is tokened rst and untokened b efore v is
tokened But vertex u can b e untokened only if all neighb ors including vertex v are
colored red see step and comment ab ove
Supp ose the algorithm terminates with all tokens placed The argumentabove
establishes that the subgraph A of H induced bytheredvertices with b oundary set
P s has pathwidth at most f t To complete the pro of we argue that in this case
the sequence of token placements establishes that A contains a sub division of B
h t
and hence must have pathwidth greater than t Since the GrowTokenTree function
only attaches p endenttokens to parenttokens we need to only to observethatthe
op eration in step sub divides the edge b etween T and its parent
Corollary Given a graph H of order n and an integer t there exists an linear
time algorithm that gives evidence that the pathwidth of H is greater than t or nds
t
a path decomp osition of width at most f t O
Pro of Weshow that program PathDecomp ositionOrSmallFatFactor runs in linear
time First if H has more than t n edges then the pathwidth of H is greater than
t By the pro of of Theorem the program terminates with either the embedded
binary tree as evidence or a path decomp osition of width at most f t
Note that the guest tree B has constant order f t and so token op erations
h t
that do not involve scanning H are constant time In function GrowTokenTree the
only non constant time op eration is the check for blue neighb ors in step While
scanning the adjacent edges of vertex u anyedgetoaredvertex can b e removed
in constant time Edge u v is also removed when step is executed Therefore
across all calls to GrowTokenTree eachedgeof H needs to b e considered at most once
for a total of O n steps In program PathDecomp ositionOrSmallFatFactor all steps
CHAPTER COMPUTING PATHWIDTH BY PEBBLING
except for GrowTokenTree are constant time The total numb er of iterations through
the lo op is b ounded by n by the termination argument following the program
The next result shows that we can improve the pathwidth algorithm by restrict
ing the guest tree This allows us to use the sub divided tree obstructions given in
Figure as guests
Corollary Any subtree of the binary tree B that has pathwidth greater
h t
than t may b e used in the algorithm for Theorem
Pro of The following simple mo di cations allow the algorithm to op erate with a
subtree The subtree is sp eci ed by a set of agged tokens in B Atworst the
h t
algorithm can p otentially emb ed all of B
h t
In step of GrowTokenTree the algorithm only lo oks for a agged untokened
child T b to place since un agged tokens need not b e placed The stopping condition
in step of PathDecomp ositionOrSmallFatFactor is changed to all agged tokens
of B are placed or so that termination occurs as soon as the subtree has
h t
b een emb edded The relab eling in step can place un agged tokens of B t on
h
vertices of H since all the ro oted subtrees of a xed height are not isomorphic If that
happ ens we expand our guest tree by adding ags with those new tokens It is easy
to see that the new guest is still a tree These newly agged tokens may b e relab eled
by future edge sub divisions that o ccur ab ove the token in the host tree Duplication
of token lab els do es not happ en if un agging is not allowed Thus the f t width
b ound is preserved
Example Using Tree from Figure sub divided K as the guest tree
of pathwidth the program trace given in Figure terminates with all vertices
colored red gray yielding a path decomp osition of width
In the pro of of Corollary one may wish to not expand the guest tree by
agging new tokens This can b e done and in fact is what wewould do in practice
Without loss of generality supp ose token T isonvertex u H and has children
that can not b e placed on H andT has one un agged sibling token T onv H
CHAPTER COMPUTING PATHWIDTH BY PEBBLING
a a a
b b b
c c c
d d d
e e e
f f f
g g g
i i i
h h h
After step
After T
After
P fa b c d eg P fa b d e f hg
a a
a
b b
b
c c
c
d d
d
e e
e
f f
f
g g
g
h i h i
h i
T after
T after
T after
P fa b g h ig
P fa b e h ig P fa b d e hg
Figure Illustrating our linear time pathwidth algorithm
CHAPTER COMPUTING PATHWIDTH BY PEBBLING
If we ignore the agging of new vertices in the current algorithm the token T would
b e removed and the parent T which has only one legitimate child would b e placed
on vertex v What happ ens to anybluevertices that are adjacenttoonlyvertex u
or its un agged subtree The answer is that they are lost and the algorithm would
not terminate unless it could emb ed the guest tree in the remaining p ortion of H
We can x this problem bychecking for un agged siblings b efore step and to shift
the token T from u to v See step below
if T had one tokened child then
replace all tokens T b S with T S
else if T P b had an un agged sibling then
replace all tokens P b S with P not b S
endif
Observe that our pathwidth algorithm provides an easy pro of of basically the
main result of BRST or its earlier variant RS that for any forest F there
is a constant c such that any graph not containing F as a minor has pathwidth at
most c
Corollary Every graph with no minor isomorphic to forest F where F is a
minor of a complete binary tree B has pathwidth at most c jB j
Pro of Without loss of generality we can run our pathwidth algorithm using as the
guest any subtree T of B that contains F as a minor Since at most jB j p ebbles
are used when wedonot ndanemb edding of T in any host graph the resulting
path decomp osition has width at most jB j
Our constant c is identical to the one given in BRST when F B They
point out that their constant c jF j is the b est p ossible since the complete graph
K with pathwidth c do es not contain any forest with c vertices
c
Further Directions
In the case that the pathwidth of an input graph G is at most k our algorithm yields
a path decomp osition that can have a width exp onential in k but that is equal in
CHAPTER COMPUTING PATHWIDTH BY PEBBLING
any case to the maximum numb er of tokens placed on the graph at anygiven time
minus It would b e interesting to know if this exp onential bad b ehavior is normal
or whether the algorithm tends to use a smaller numb er of tokens in practice Since
the p ebbling pro ceeds according to a greedy strategy with much exibility there may
b e placement heuristics that can improve its p erformance on typical instances
Chapter
Conclusion
This chapter summarizes the results of this dissertation and then concludes with
two imp ortant applications that concern forbidden substructure characterization of
graphs
A Summary of the Main Results
Our main computational achievements given in Part I I of this dissertation show
that computing obstructions for simple minor order lower ideals is feasible for small
pathwidth Here wedevelop ed a computational theory given in Part I that extends
the original Fellows Langston approach for computing obstruction sets For a wide
range of targeted graph families in fact all b eing parameterized lower ideals Ta
ble summarizes the largest pathwidth b ounds that our software can currently
explore Our exp erimental research has shown that these pathwidth b ounds may in
crease by one if universal distinguisher searching is available see Sections and
Furthermore with new theory b eing develop ed there is ample opp ortunityto
characterize additional lower ideals with our basic metho d We exp ect that for any
lower ideal the corresp onding treewidth b ounds regarding achievable obstruction set
computations will b e equivalent to the largest feasible pathwidth b ounds That is
wehave no evidence to suggest that our treewidth enumeration metho ds b ecome impractical at smaller widths
CHAPTER CONCLUSION
Table Tractability b ounds for various parameterized lower ideals
Minor Order Lower Ideal Largest Pathwidth
k Edge Bounded IndSet
Vertex Cover
Feedback Edge Set
Path Cover
Path Cover
Cycle Cover
Feedback Vertex Set
Outer Planar
Planar
Genus torus
Pathwidth
Wenow list a few noteworthy theory results that were presented in Part I of this
dissertation
Intro duced practical t parse enumeration schemes based on canonic representa
tions for graphs of b ounded pathwidth and treewidth
Presented a small search space technique of b ounded combinatorial width for
computing minor order obstructions Our Pre x Lemma establishes a search
tree of minimal t parses as growable no des guaranteeing a search that termi
nates
Develop ed su cient rules for computing disconnected minor order obstructions
for certain parameterized lower ideals The exp ected growth rate on the number
of obstructions for parameterized lower ideals was also studied
Utilized a distributive programming approach across many hardware platforms for obstruction set computations
CHAPTER CONCLUSION
Besides characterizing many avors of graph families by obstruction sets in Part
I I of this dissertation weprovided several other useful results such as the following
family sp eci c items for input graphs of b ounded pathwidth t parses
An optimal nite state algorithm for k Vertex Cover including a linear time
vertex cover algorithm
A nite state dynamic program for k Feedback Vertex Set including a lin
ear time feedbackvertex set algorithm
A testset for b oth the k Feedback Vertex Set and k Feedback Edge Set
graph families and a nonminimal pretest for k Feedback Edge Set
A linear time dynamic program to determine the maximum path length and a
congruence for k Path Cover p
A testset for k Cycle Cover and some testsets for various families of
graphs that have b ounded path or cycle lengths
A linear time dynamic program for Outer Planar and the minimum K
cover
Lastly inPart I I I of this dissertation wecontributed the following applied results
Equated a VLSI layout problem k CVS with a computational biology problem
k ICG
Showed that nding the minimum size testset for a minimal automaton or
canonical congruence is NP hard
Develop ed a simple linear time algorithm that for xed k determines if a graph
has pathwidth greater than k or nds a path decomp osition of width at most
k
O
CHAPTER CONCLUSION
Two Key Applications
We conclude this dissertation by mentioning two areas of research that are imp or
tant consequences of our work These two applications are of interest to b oth the
theoretical and industrial computer science communities
First wehave empiricallyshown that mathematical theorem proving sp eci cally
for the case of nding substructure characterizations of minor order lower ideals can
b e automated Wehave presented a natural and easy way to nd obstruction sets
for those characterizable families whose obstructions are b ounded by some constant
combinatorial width The Graph Minor Theorem guarantees a nite numb er of ob
structions and hence a b ound on b oth the pathwidth and treewidth Our metho d
requires two additional ingredients consisting of a memb ership algorithm and a re ne
ment of the canonical congruence for the lower ideal This dissertation has shown that
these required congruences are easy to pro duce We demonstrated with several ex
amples two design techniques for obtaining these family sp eci c congruences namely
by testsets of t b oundaried graphs and nite state dynamic programs for t parses
For an application with real world p otential wehavedevelop ed a to ol that pro
vides graph algorithm designers with the following opp ortunity As mentioned ear
lier in Section one can take a partial set of obstructions for any minor order
lower ideal F and build an approximating automaton for recognizing those graphs of
b ounded width within F We can do this for an y desired pathwidth or treewidth A
randomized feature of our software allows one to generate partial sets of obstructions
without the need for a nite index family congruence for F thereby allowing the
non sp ecialist to takeadvantage of our system That is wecannow implementa useful graph algorithm compiler
Annotated Bibliography
ACP S Arnb org D G Corneil and A Proskurowski Complexity of nding em
b eddings in a k tree SIAM Journal on Algebraic Discrete Methods
April
Contains the rst p olynomial time algorithm for determining the treewidth of a
graph for xed k Also the partial k tree problem is shown to b e NP complete
Provides a listing of the treewidth three obstructions
ACPS S Arnb org D G Corneil A Proskurowski and D Seese An algebraic theory
of graph reduction In Proceedings of the Fourth Workshop on Graph Gram
mars and Their Applications to Computer Science volume of LectureNotes
on Computer Science pages Springer Verlag To app ear Journal
of the Association Computing Machinery
Shows how memb ership in classes of graphs de nable in monadic second order
logic and of b ounded treewidth can b e decided by nite sets of terminating reduc
tion rules
AF K Abrahamson and M Fellows Finite automata b ounded treewidth and
well quasiordering In N Rob ertson and P D Seymour editors Graph Stru
tureTheory volume of Contemporary Mathematics pages
Formally known as Cutset regularity b eats well quasi ordering for b ounded
treewidth
Contains a necessa ry and su cient condition cutset regularity for a family of
graphs to b e recognizable from structural parse trees by nite state tree automata
AH D S Archdeacon and P Huneke On cubic graphs which are irreducible for
nonorientable surfaces Journal of Combinatorial Theory Series B
ALS S Arnb org J Lagergren and D Seese Easy problems for tree decomp osable
graphs Journal of Algorithms Also see extended abstract
in volume of Lecture Notes on Computer Science
Mono dic second order logic is presented to showthatalarge set of graph families
is nite state This is an alternate pro of of Courcelle s result
AP S Arnb org and A Proskurowski Characterization and recognition of partial
trees SIAM Journal on Algebraic Discrete Methods April
A set of con uent graph reductions is given such that a graph can b e reduced to
the empty graph if and only if it is a subgraph of a tree
AP S Arnb org and A Proskurowski Linear algorithms for NP hard problems
restricted to partial k trees Discrete Applied Mathematics pages
AP S Arnb org and A Proskurowski Canonical representations of partial and
trees BIT
APC S Arnb org A Proskurowski and D Corneil Minimal forbidden minor charac
terization of a class of graphs Col loquia MathematicaSocietatis J anos Bolyai
A graph is a partial tree if and only if it do es not have a minor isomorphic to
any of four obstructions
APS S Arnb org A Proskurowski and D Seese Forbidden minors characterization
of partial trees Discrete Mathematics
APS S Arnb org A Proskurowski and D Seese Monadic second order logic tree
automata and forbidden minors In Proceedings of the th Workshop on Com
puter ScienceLogic CSL volume of Lecture Notes on Computer Sci
ence pages Springer Verlag
Arc D Archdeacon A Kuratowski Theorem for the projective plane Ph D dis
sertation The Ohio State University Also see Journal of Graph Theory
The authorsketches a pro of that the list of irreducible graphs for the projec
tive plane is complete Also it is p ointed that there are at least irreducible
graphs for the Klein b ottle
Arc D Archdeacon The complexity of the graph emb edding problem In R Bo
dendiek and R Henn editors Topics in Combinatorics and Graph The ory
pages Physica Verlag
Arc D Archdeacon private communication Dept of Mathematics and
Statistics UniversityofVermont
Arn S Arnb org Reduced state enumeration another algorithm for reliabilityeval
uation IEEE Transactions on Reliability
Arn S Arnb org E cient algorithms for combinatorial problems on graphs with
b ounded decomp osability A survey BIT Invited pap er
This is a go o d survey pap er on table based reduction metho ds for graphs with
b ounded treewidth
Arn S Arnb org Graph decomp ositions and tree automata in reasoning with un
certainty JournalofExperimental and Theoretical AI to app ear
AST N Alon P D Seymour and R Thomas A separator theorem for graphs with
an excluded minor and its applications In Proceedings of the ACM Symposium
on the Theory of Computing volume pages Baltimore
Maryland
Supplies extensions of the many known applications of the Lipton Tarjan separator
theorem for the class of planar graphs to any class of graphs with an excluded
minor
BC M Bauderon and B Courcelle Graph expressions and graph rewritings Math
ematical System Theory
The notion of a context free graph grammar is intro duced The notion of an equa
tional set of graphs follows from a de ned algebraic structure A notion of graph
rewriting is also given based on a categorical approach
BD D Biensto ck and N Dean Some results on minimum face covers in planar
graphs February Bell Communications Research Morristown New Jer
sey
This pap er discusses some obstructions to the existence of small face covers
BdF H L Bo dlaender and B L E de Fluiter Intervalizing k colored graphs
Technical rep ort UU CS Department of Computer Science Utrecht
University Utrecht the Netherlands Extended abstract to app ear in
pro ceedings ICALP
Solves the complexity issues regarding k ICG problem
BFH H L Bo dlaender M R Fellows and M T Hallett Beyond NP completeness
for problems of b ounded width Hardness for the W hierarchy extended ab
stract In Proceedings of the ACM Symposium on the Theory of Computing
pages
BFL D J Brown M R Fellows and M A Langston Polynomial time self
reducibility Theoretical motivations and practical results International Jour
nal of Computer Mathematics
The notion of self reducibility is explained for tackling the promised p olynomial
time decision algorithms which are only known via nonconstructive means
BFP J C Bermond P Fraigniaud and J G Peters Antep enultimate broadcasting
Netorks
This pap er uses path covers in the design of minimum broadcast graphs
BGHK H L Bo dlaender J R Gilb ert H Hafsteinsson and T Kloks Approximating
treewidth pathwidth and minimum elimination tree height Technical Rep ort
RUU CS Dept of Computer Science University of Utrecht P O Box
TAUtrecht the Netherlands January To app ear Journal
of Algorithms
Vertex separators are used to approximate width problems
BH T Beyer and S M Hedetniemi Constant time generation of ro oted trees
SIAM Journal on Computing
BHKY J Battle F Harary Y Kadama andJ Youngs Additivity of the genus of a
graph Bul letin of the American Mathematical Society
Shows that the genus of a graph equals the sum of the genus of the individual
blo cks
Bie D Biensto ck Graph searching path width tree width and related problems a
survey In Reliability of Computer and Communication Networks volume of
DIMACS Series in Discrete Mathematics and Theoretical Computer Science
pages Asso ciation for Computing Machinery
BJM T Beyer W Jones and S L Mitchell Linear algorithms for isomorphism of
maximal outerplanar graphs Journal of the Association Computing Machinery
BK H L Bo dlaender and T Kloks Better algorithms for pathwidth and treewidth
of graphs In Proceedings of the International Col loquium on Automata Lan
guages and Programming volume of Lecture Notes on Computer Science
pages Springer Verlag th ICALP
For all constants k an explicit O n log n algorithm is given to decide whether the
treewidth pathwidth of a graph is at most k and if so nds the decomp ositions
BK H L Bo dlaender and T Kloks Approximating treewidth and pathwidth of
some classes of p erfect graphs In Proceedings of the rd International Sympo
sium on Algorithms and Computation ISAAC volume of LectureNotes
on Computer Science pages Springer Verlag
BK H L Bo dlaender and T Kloks E cient and constructive algorithms for the
pathwidth and treewidth of graphs preprint
BL D Biensto ck and M A Langston Algorithmic implications of the graph mi
nor theorem to app ear Handbook of Operations Research and Management
Science Volume on Networks and Distribution
BLW M W Bern E L Lawler andA L Wong Wh y certain subgraph computa
tions require only line time In IEEE Symposium on Foundations of Computer
ScienceProceedings pages th FOCS Portland OR
BLW M W Bern E L Lawler and A L Wong Linear time computation of optimal
subgraphs of decomp osable graphs Journal of Algorithms
Full version of FOCS pap er
For the class of graphs with b ounded combinatorial width a dynamic programming
approach is used for general subgraph problems
BM J Bondy and U Murty Graph Theory with Applications MacMillan
BM D Biensto ck and C L Monma On the complexityofcovering vertices by
faces in a planar graph SIAM Journal on Computing
An algorithm is give which either determines if a graph is not k planaror generates
k
an appropriate emb edding and asso ciated minimum cover in O c n time where
c is a constant
BM H Bo dlaender and R M ohring The pathwidth and treewidth of cographs
SIAM Journal on Discrete Mathematics
Shows that the pathwidth equals the treewidth for cographs and gives a linear time
algorithm for nding a tree decomp osition
Bo d a H L Bo dlaender Classes of graphs with b ounded tree width Technical Rep ort
RUU CS Dept of Computer Science University of Utrecht P O Box
TAUtrecht the Netherlands Decemb er Also in Bul letin
of the EATCS
A numb er of classes of graphs are shown to b e sub classes of the graphs with
tree width b ounded by some constant k
Bo d b H L Bo dlaender Planar graphs with b ounded tree width Technical rep ort
Dept of Computer Science University of Utrecht P O Box TA
Utrecht the Netherlands
Discusses k outerplanar graphs planar graphs with b ounded radius and Halin graphs
Bo d b H L Bo dlaender Some classes of graphs with b ounded treewidth Bul letin of
the EATCS
Bo d a H L Bo dlaender Dynamic programming algorithms on graphs with b ounded
treewidth In Proceedings of the International Col loquium on Automata Lan
guages and Programming volume of Lecture Notes on Computer Science
pages Springer Verlag th ICALP
Gives an O n time algorithm forthe k disjoint cycles problem
Bo d H L Bo dlaender Improved self reduction algorithms for graphs with b ounded
treewidth In Graph Theoretic Concepts in Computer Science volume of
Lecture Notes on Computer Science pages Springer Verlag
Also in Discrete Applied Mathematics
Self reduction is used to construct solutions to many of the Rob ertson Seymour
nonconstructive p olynomial time families
Bo d H L Bo dlaender Polynomial algorithms for graph isomorphism and chromatic
index on partial k trees Journal of Algorithms
Two di erent approaches are used to show that the Chromatic Index and Graph
Isomorphism problems are solveable in p olynomial time when restricted to the class
of graphs with treewidth k
Bo d H L Bo dlaender On disjoint cycles In Proc eedings of the th International
Workshop on Graph Theoretic Concepts in Computer Science WG volume
of Lecture Notes on Computer Science pages Springer Verlag
Also UniversityofUtrechttechnical rep ort RUU CS
Gives an O n k FVS algorithm
Bo d c H L Bo dlaender A tourist guide through treewidth Acta Cybernetica
A short overview of algorithmic graph theory that deal with the notions of
treewidth and pathwidth
Bo d b H L Bo dlaender On linear time minor tests and depth rst search Journal of
Algorithms Also in volume of Lecture Notes on Computer
Science
Presents e cient minor tests when the set of graphs has a k grid as a minor
k
and a circus graph as a minor Also gives an O k n time algorithm to determine
whether a graph has a cycle or path or length k if it exists
Bo d a H L Bo dlaender A linear time algorithm for nding tree decomp ositions of
small treewidth In Proceedings of the ACM Symposium on the Theory of Com
puting volume
This is the end of the st o ry except p erhaps nding a practical algorithm regard
ing the k parameterized treewidth problem As a consequence every minor closed
class of graphs that do es not contain all planar graphs has a linear time recognition
algorithm
Bo d H L Bo dlaender A partial k arb oretum of graphs with b ounded treewidth
Preliminary manuscript Dept of Computer Science University of Utrecht
P O Box TA Utrecht the Netherlands Octob er
Bon J A Bondy Pancyclic graphs recent results pages North Holland
Amsterdam Collo quia Mathematica So cietatis J anos Bolyai volume
Halin graphs are intro duced
Bor R B Borie Recursively ConstructedGraph Families Membership and Linear
Algorithms Ph D thesis Georgia Institute of Technology Scho ol of Informa
tion and Computer Science
This thesis uses t terminal graphs b oundaried graphs with recursive rules for
graph families e g partial k trees and an automated technique for generating
linearalgorithms from a predicate calculus
BP L Beineke and R Pipp ert Prop erties and characterizations of k trees Math
ematika
BPT R Borie R Parker and C Tovey The regular forbidden minors of partial
trees Technical rep ort Scho ol of ISYE Georgia Institute of Technology
Tech Rep ort No J
BPT R B Borie R G Parker and C A Tovey Deterministic decomp osition of
recursive graph classes SIAM Journal on Discrete Mathematics
Examines how the series parallel graphs can b e generalized i e with recursive
rules
BRST D Biensto ck N Rob ertson P D Seymour and R Thomas Quickly excluding
a forest Journal of Combinatorial The ory Series B
CB C Colb ourn and K Bo oth Linear time automorphism algorithms for trees
interval graphs and planar graphs SIAM Journal on Computing
February
CD K Cattell and M J Dinneen A characterization of graphs with vertex cover
up to ve In V Bouchitte and M Morvan editors Orders Algorithms and
Applications ORDAL volume of Lecture Notes on Computer Science
pages Springer Verlag July
Contains a previous version of Chapter
CDDF K Cattell M J Dinneen R G Downey and M R Fellows Computational
asp ects of the graph minor theorem Obstructions for unions and intertwines
University of Victoria working manuscript
CDF a K Cattell M J Dinneen and M R Fellows Obstructions to within a few ver
tices or edges of acyclic In Proceedings of the Fourth Workshop on Algorithms
and Data Structures WADS volume of Lecture Notes on Computer
Science pages Springer Verlag August
This pap er is an extended abstract version of Chapter
CDF b K Cattell M J Dinneen and M R Fellows A simple linear time algorithm
for nding path decomp ositions of small width Information Processing Letters
to app ear
This pap er is almost identical to Chapter
CG J H Conway and C Gordon Knots and links in spatial graphs Journal of
Graph The ory
CG J Chen and J L Gross Kuratowski typ e theorems for average genus Journal
of Combinatorial Theory Series B
The authors de ne average genus of a graph and provide obstructions forsmall
fractional averages up to c They ask if there is a way to implement this
metho d systematically for fractions up to c or
CKK V Chvatal D A Klarner and D E Knuth Selected combinatorial research
problems Technical rep ort Dept of Computer Science Stanford University
June
The pap er characterizes unit distance graphs bytwoforbidden subgraphs K and
K
CL G Chartrand and L Lesniak Graphs and Digraphs Wadsworth Inc
CM B Courcelle and M Mosbah Monadic second order evaluations on tree
decomp osable graphs Theoretical Computer Science
Presents graphs as logical structures almost like our t parse op erator sets
Cou a B Courcelle The monadic second order logic of graphs I Recognizable sets of
nite graphs Information and Computation
Cou B Courcelle The monadic second order logic of graphs I I De nable sets of
in nite graphs Mathematical Systems Theory
Cou B Courcelle The monadic second order logic of graphs I I I treewidth forbid
den minors and complexity issues Informatique Th eorique
Cou b B Courcelle The monadic second order logic of graphs IV Every equational
graph is de nable Annals of Pure and AppliedLogic
Cou B Courcelle The monadic second order logic of graphs V On closing the
gap b etween de nability and recognizability Theoretical Computer Science
Cou B Courcelle The monadic second order logic of graphs VI On several rep
resentations of graphs by related structures Discrete Applied Mathematics
Cou b B Courcelle The monadic second order logic of graphs VI I Graphs as rela
tional structures Theoretical Computer Science
COS D G Corneil S Olariu and L Stewart Computing a dominating pair in an
asteroidal triple free graph in linear time In Proceedings of the Fourth Work
shop on Algorithms and Data Structures WADS volume of Lecture
Notes on Computer Science pages Springer Verlag August
Cou B Courcelle A representation of graph by algebraic expressions and its use for
graph rewriting systems In Proceedings of the rd International Workshop on
Graph Grammers volume of Lecture Notes on Computer Science pages
Springer Verlag
Cou a B Courcelle Graph grammars monadic second order logic and the theory of
Bul letin of the EATCS Also in Contemporary graph minors
Mathematics
This is a survey of the relationships b etween the descriptions of sets of graphs
byformulas of monadic second order logic by context free hyp eredge and vertex
replacement graph grammarsandbyforbidden minors
CS F R K Chung and P D Seymour Graphs with small band width and cut
width Discrete Mathematics
CSTV R Cohen S Sairam R Tamassia and J S Vitter Dynamic algorithms
for b ounded tree width graphs Technical rep ort Dept of Computer Science
Brown University Tech Rep ort CS
CV D Copp ersmith and U Vishkin Solving NP hard problems in almost trees
Vertex cover Discrete Applied Mathematics
The authors present a linear time algorithm for determining the vertex cover for
graphs that are parameterized bytwoparameters a the numb er of edges more
than a tree and k the maximum a over all biconnected comp onents of the graph
Dec R W Decker The Genus of Certain Graphs Ph D dissertation The Ohio
State University
The concept of irreducibile graphs forthetorus is investigated partial list is
given
Deo N Deo Graph Theory with Applications to Engineering and Computer Science
Prentice Hall
DF R Downey and M R F ellows Parameterized computational feasibility In
P Clote and J Remmel editors Feasible Mathematics II pages
Birkhauser Pro ceedings of the nd Cornell Workshop on Feasible Math
ematics
Dis Special issue E cient Algorithms and partial k trees volume of Discrete
Applied Mathematics Octob er numb ers
DR I S Du and J K Reid The multifrontal solution of inde nite sparse symmet
ric linear equations ACM Transactions on Mathematical Software
DR H Didjev and J Reif An e cient algorithm for the genus problem with explicit
construction of forbidden subgraphs In IEEE Symposium on Foundations of
Computer ScienceProceedings volume pages
Gives some order b ounds for genus k obstructions
DS W W M Dai and M Sato Minimal forbidden minor characterization of
planar partial trees and applications to circuit layout IEEE International
Symposium on Circuits and Systems pages
Duf R J Du n Top ology of series parallel networks Journal of Mathematical
Analysis and Applications
Go o d motivation of these network graphs with regards to the resistance rules of
re precisely series parallel graphs Ohm De nes con uent graphs and shows they a
EMC E El Mallah and C Colb ourn Partial k tree algorithms Congressus Numer
antium
EMC E El Mallah and C Colb ourn On two dual classes of planar graphs Discrete
Mathematics
Gives a characterization of planarpartial trees in terms of two obstructions
Epp D Eppstein Parallel recognition of series parallel graphs Information and
Computing May
EST J A Ellis I H Sudb orough and J Turner Graph separation and search
numb er Rep ort DCS IR Dept of Computer Science University of Victo
ria P O Box Victoria B C Canada V W P August
Contains the rst p olynomial time algorithm for determining if a graph has path
width k or less and a fast linear time algorithm for determining the pathwidth of
trees
EST J Ellis I H Sudb orough and J Turner The vertex separation and search
numb er of a graph Information and Computation
FBS D Fern andez Baca and G Slutzki Solving parametric problems on trees
Journal of Algorithms
FBS D Fern andez Baca and G Slutzki Parametric problems on graphs of b ounded
treewidth In Proceedings of the rdScandinavian Workshop on Algorithm
Theory volume of Lecture Notes on Computer Science pages
Springer Verlag
FBS D Fern andez Baca and G Slutzki Optimal parametric problems on graphs
of b ounded treewidth In Proc eedings of the rdScandinavian Workshop on
Algorithm Theory volume of Lecture Notes on Computer Science pages
Springer Verlag
Fel M R Fellows private communication Dept of Computer Science University
of Victoria
Fel M R Fellows The Rob ertson Seymour theorems A survey of applications
Contemporary Mathematics
Applications of the Rob ertson Seymour theorems to a varietyofproblems in con
crete computational complexityare surveyed
FG P A Firby and C F Gardiner SurfaceTopology volume of Mathematics
and its Applications Ellis Horwo o d second edition
FHW M R Fellows M T Hallett and H T Wareham DNA physical mapping
Three ways di cult In T Lengauer editor Proceedings of European Sym
posium on Algorithms ESA volume of Lecture Notes in Computer
Science pages Springer Verlag Berlin
wn to The combinatorial problem of Intervalizing Colored Graphs or ICG is sho
b e intractable in three di erent ways it is NP complete it is hard for
the parameterized complexity class W and it is not nite state for b ounded
treewidth or pathwidth
Fis P C Fishburn Interval Orders and Interval Graphs A Study of Partial ly
Ordered Sets John Wiley NewYork
FKL M R Fellows N G Kinnersley and M A Langston Finite basis theorems
and a computation integrated approach to obstruction set isolation Extended
abstract Dept of Computer Science Washington State University Pullman
WA Novemb er Also in Pro ceedings of Computers and Math
ematics Conference
FKMP M Fellows J Krato chv M Middendorf and F Pfei er Induced minors and
related problems Algorithmica
Explores the similarities and di erences of minors and induced minors Combina
torial problems with restricted input to planargraphsare also considered
FL M R Fellows and M A Langston Nonconstructiveadvances in p olynomial
time complexity Information Processing Letters Octob er
The connection b etween the Rob ertson Seymour p osets and the Gate Matrix
Layout problem is given
FL a M R Fellows and M A Langston Layout p ermutation problems and well
partially ordered sets In Proceedings of the th MIT ConferenceonAdvanced
Research in VLSI Cambridge Mass Mar pages MIT press
Low degree p olynomial time algorithms for a numb er of well studied VLSI layout
problems along with self reducibility results a re given
FL b M R Fellows and M A Langston Nonconstructive to ols for proving
p olynomial time decidabili ty Journal of the Association Computing Machin
ery
De nes what Rob ertson Seymour p osets are and shows how to use these to ols
for determining the complexity of di cult combinatorial decision problems
FL a M R Fellows and M A Langston An analogue of the Myhill Nero de Theorem
and its use in computing nite basis characterizations In IEEE Symposium
on Foundations of Computer ScienceProceedings volume pages
Using a graph theoretic analogue of the Myhill Nero de characterization of regular
languages this pap er establishes that for many applications obstruction sets are
computable by known algorithms
FL b M R Fellows and M A Langston On search descision and the e ciency
of p olynomial time algorithms In Proceedings of the ACM Symposium on the
Theory of Computing volume pages Also in Journal of
Computer and System Sciences
Discusses the problem of equating tractability with p olynomial time decidability
in light of the recent advances in well partial order theory esp ecially the seminal
contributions of Rob ertson and Seymour
FL M R Fellows and M A Langston On well partial order theory and its appli
cation to combinatorial problems of VLSI design SIAM Journal on Discrete
Mathematics February
Lo w degree p olynomial time algorithms are nonconstructively shown for the rst
time formanywell studied graph layout placement and routing problems
FMR I S Filotti G L Miller and J Reif On determining the genus of a graph in
O g
O v steps In Proceedings of the ACM Symposium on the Theory of Com
puting volume pages Atlanta Georgia Preliminary Rep ort
First p olynomial time algorithms are presented for testing for xed genus g
FRS H Friedman N Rob ertson and P D Seymour The meta mathematics of the
graph minor theorem In Applications of Logic to Combinatorics volume of
Contemporary Mathematics pages American Mathematical So ciety
Gav F Gavril The intersection of graphs of subtrees in trees are exactly the chordal
graphs Journal of Combinatorial Theory
GGJK M R Garey R L Graham D S Johnson and D E Knuth Complexity
results for bandwidth minimization SIAM Journal on Applied Mathematics
May
Bandwidth is NP complete even when restricted to free trees with maximum
degree three
GGKS P W Goldb erg M C Golumbic H Kaplan and R Shamir Four strikes
against physical mapping of DNA Technical rep ort The Moise and
Frida Eskenasy Institute of Computer Sciences Tel Aviv University December
Four simpli ed mo dels of the DNA fragment reconstruction p roblem lead to NP
complete decision problems In particular a simple NP completeness pro of is given
regarding the pathwidth of a bipartite graph
GH H H Glover and J P Huneke There are nitely many Kuratowski graphs
for the pro jective plane In Graph Theory and RelatedTopics pages
Academic Press Inc
The pap er shows that a graph emb eds in the projective plane if and only if it do es
not contain a subgraph homeomorphic to one of a nite list of graphs This is done
indep enden t of any detailed list of graphs
GHW a H H Glover J P Huneke and C S Wang graphs which are irreducible
for the pro jective plane Journal of Combinatorial Theory Series B
GHW b H H Glover J P Huneke and C S Wang On a Kuratowski theorem for
the pro jective plane In Graph Theory and RelatedTopics pages
Academic Press Inc
The projective plane can b e systematically characterized from thirty ve particular
irreducible graphs An app endix contains drawings of all such obstructions
GJ M R Garey and D S Johnson Computers and Intractability A Guide to the
Theory of NP completeness W H Freeman and Company
The b est intro duction to the theory of NP completeness and a catalog of classical
NP complete problems
GKS M C Golumbic H Kaplan and R Shamir Graph sandwic h problems Tech
nical rep ort The Moise and Frida Eskenasy Institute of Computer
Sciences Tel Aviv University December
The graph sandwich problem forprop erty X is de ned Many NP completeness
pro ofs are given for comparability graphs p ermutation graphs and several other
families
GKS M C Golumbic H Kaplan and R Shamir On the complexity of DNA
physical mapping Technical rep ort The Moise and Frida Eskenasy
Institute of Computer Sciences Tel Aviv University January to app ear
Advances in Applied Mathematics
They indep ende ntly de ned the k ICG problem under the name Colored Interval
Sandwich Problem and showittobe NP complete by a reduction that do es not
imply W hardness
GLR a R Govindan M A Langston and S Ramachandramurthi Cutwidth approxi
mation in linear time Technical rep ort Dept of Computer Science University
of Tennessee Knoxville TN
An algorithm to determine whether K is immersed in a graph is given
GLR b R Govindan M A Langston and S Ramachandramurthi A practical ap
proachtolayout optimization Technical rep ort Dept of Computer Science
UniversityofTennessee Knoxville TN
Presents fast layout algorithms generated by using partial obstruction sets
GN A Gupta and N Nishimura Sequential and parallel algorithms for emb edding
problems on classes of partial k trees In Pr oceedings of the rdScandinavian
Workshop on Algorithm Theory volume of Lecture Notes on Computer
Science pages July
Includes subgraph isomorphism and top ological emb edding algorithms known to
be NP complete for general partial k trees
GSK D Granot and D Skorin Kap ov On some optimization problems on k trees
and partial k trees Discrete Applied Mathematics
GS E M Gurari and I H Sudb orough Improved dynamic programming algo
rithms for bandwidth minimization and the MinCut linear arrangement prob
lem Journal of Algorithms
k
Bandwidth and cutwidth for xed k can b e solved in O n
GS D Grinstead and P Slater A recurrence template for several parameters in
series parallel graphs Discrete Applied Mathematics
GSV Y Gurevich L Sto ckmeyer and U Vishkin Soving NP hard problems on
graphs that are almost trees and an application to facility lo cation problems
Journal of the Association Computing Machinery
GT M R Garey and R E Tarjan A linear time algorithm for nding all feedback
vertices Information Processing Letters Octob er
The autho rs present an O m time algorithm which uses depth rst search to nd
all feedback vertices in an arbitrary strongly connected graph And generalized for
k
xed k the algorithm nds feedback sets of size k in time O n m
Gup A Gupta Constructive Issues in Tree Minors Ph D thesis Universityof
Toronto
Gives a constructive pro of that trees are well quasi ordered under the minoror
dering and shows that nite automata are su cient for recognizing minor closed
tree families
Har F Harary Graph Theory Addison Wesley Mass
He X He E cient parallel algorithms for series parallel graphs Journal of Algo
rithms
Contains algorithms for coloring and depth rst breadth rst spanning trees
HHL E Hare S Hedetniemi R Laskar K Peters and T Wimer Linear time com
putabilityofcombinatorial problems on generalized series parall el graphs In
D S Johnson T Nishizeki A Nozaki and H S Wilf editors Perspectives in
Computing volume pages Academic Press Inc Pro ceed
ings of the Japan U S Joint Seminar June Kyoto Japan
Illustrates an emerging theory metho dology for constructing linear time graph al
gorithms byproviding such algorithms for nding the maximum cut and the maxi
mum cardinality of a minimal dominating set for a generalized series parallel graph
HM J P Hutchinson and G L Miller On deleting vertices to make a graph of
p ositivegenus planar Prospectives in Computing
HR N Harts eld and G Ringel Pearls in Graph Theory A Comprehensive Intro
duction Academic Press Inc
HT J E Hop croft and R E Tarjan E cient planarity testing Journal of Algo
rithms
The classic rst known linear time algorithm for testing planarityispresented
here
HT R Hassin and A Tamir E cient algorithms for optimization and selection
on series parallel graphs SIAM Journal on Algebraic Discrete Methods
HU J E Hop croft and J D Ullman Introduction to Automata Theory Addison
Wesley Mass Formerly titled Formal Languages and their Relation to
Automata
HY X He and Y Yesha Binary tree algebraic computation and parallel algorithms
for simple graphs Journal of Algorithms
Presents dynamic programming techniques forseriesparallel graph and other re
lated families
IR A Itai and M Ro deh Finding a minimum circuit in a graph In Proceedings
of the ACM Symposium on the Theory of Computing pages th
STOC
Gives an O n algorithm which nds either the shortest cycle in a graph or
shortest plus one
Joh D S Johnson The NP completeness column An ongoing guide Journal of
Algorithms th edition reduced graph family instances
Joh D S Johnson The NP completeness column An ongoing guide Journal of
Algorithms th edition The many faces of p olynomial time
Discusses Rob ertson Seymour subgraph homeomorphism and minor containment
problems as well as nonconstructive pro ofs
Kar A Karab eg Classi cation and detection of obstructions to planarity Linear
and Multilinear Algebra
KGS D Kaller A Gupta and T Shermer Regular factors in the complements
of partial k trees In Proceedings of the Fourth Workshop on Algorithms and
Data Structures WADS volume of Lecture Notes on Computer Science
pages Springer Verlag August
Kin N G Kinnersley The vertex separation numb er of a graph equals its path
width Information Processing Letters
Contains a simple pro of showing the equivalence b etween the pathwidth and a
VLSI linearlayout problem
Kin N G Kinnersley Obstruction set isolation for layout permutation problems
Ph D Thesis Dept of Computer Science Washington State University Pull
man WA
Kin N G Kinnersley Constructive obstruction set isolation for the Min Cut Linear
Arrangement Tec hnical Rep ort TR Dept of Computer Science Univer
sity of Kansas January
KIU Y Ka jitani A Ishizuka and S Ueno Characterization of partial trees in
terms of structures Graphs and Combinatorics
KK a N G Kinnersley and W M Kinnersley An e cient p olynomial time algorithm
for three track gate matrix layout The Computer Journal
KK b N G Kinnersley and W M Kinnersley Tree automata for cutwidth recog
nition Technical Rep ort TR Dept of Computer Science Universityof
Kansas January
KL N G Kinnersley and M A Langston Obstruction set isolation for the Gate
Matrix Layout problem Discrete Applied Mathematics
Klo a T Kloks Treewidth Ph D dissertation Dept of Computer Science Utrecht
University P O Box TA Utrecht the Netherlands
Among many results this dissertation gives an algorithm to determine if a graph
is pathwidth or treewidth k when given a decomp osition of any width greater
than k
Klo b T Kloks In Treewidth computations and approximations volume of
Lecture Notes on Computer Science Springer Verlag
This is a more available source than Kloks Ph D dissertation
KM A K ezdy and P McGuinnes Sequential and parallel algorithms to nd a K
minor Discrete Applied Mathematics
An O n sequential algorithm is presented that given a graph either returns a
K minororreports that no such minor exists
Koz D Kozen On the Myhill Nero de Theorem for trees Bul letin of the EATCS
KP L M Kirousis and C H Papadimitriou Searching and p ebbling Theoretical
Computer Science
KPS M Khalifat T Politof and A Satyanarayana On minors of graphs with at
least n edges Journal of Graph Theory
Kra D Kratsch Finding the minimum bandwidth of an interval graph Informa
tion and Computation
Gives an O n algorithm for determining the bandwidth of interval graphs al
though it is known to b e NP complete for catepillers with hairs of length at most
and forbinary trees
Kru J B Kruskal Well quasi ordering the tree theorem and Vaszsonyi s conjec
ture Transactions of American Mathematical Society
Kru J B Kruskal The theory of well quasi ordering a frequently rediscovered
concept Journal of Combinatorial Theory Series A
KS H Kaplan and R Shamir Pathwidth bandwidth and completion problems to
prop er interval graphs Technical rep ort The Moise and Frida Eskenasy
Institute of Computer Sciences Tel Aviv Univ ersity November
Twoproblems motivated by molecular biology are studied Both are shown to b e
p olynomial for xed k but in general one is NP hard while the other is hard for
W
KST H Kaplan R Shamir and R E Targan Tractability of parameterized com
pletion problems on chordal and interval graphs minimum ll in and physical
mapping In Proceedings of the th Annual IEEE Conference on the Founda
tions of Computer Science pages
KT A Kornai and Z Tuza Narrowness pathwidth and their application in natural
language pro cessing Discrete Applied Mathematics
Points out the imp ortance of graphs with pathwidth in connection with natural
language pro cessing
Kur K Kuratowski Sur le probl e des courb es gauches en top ologie Fundamental
Mathematics
KyK T Kikuno N yoshida and Y Kakuda A linear algorithm for the domination
numb er of series parallel graphs Discrete Applied Mathematics
LA J Lagergren and S Arnb org Finding minimal forbidden minors using a nite
congruence In Proceedings of the International Col loquium on Automata Lan
guages and Programming volume of Lecture Notes on Computer Science
pages Springer Verlag th ICALP
Using an algebra of i sourced graphs an e ective way to compute the minimal
forbidden minors of graphs of b ounded treeidth is derived from nite congruences
that recognizes minor closed families Also an algorithm that recognizes graphs of
treewidth at most k that runs in lineartimeispresented
Lag J Lagergren E cient parallel algorithms for tree decomp osition and related
problems In IEEE Symposium on Foundations of Computer ScienceProceed
ings pages th FOCS
Lag J Lagergren Algorithms and Minimal Forbidden Minors for Tree decomposible
Graphs Ph D dissertation Royal Institute of Technology Stockholm Sweden
March Dept of Numerical Analysis and Computing Sciences
Lag J Lagergren An upp er b ound on the size of an obstruction In N Rob ertson
and P D Seymour editors Graph StructureTheory volume of Contem
porary Mathematics pages
The authorproves constructively that every minor closed family which is recog
nized by a nite congruence and has an obstruction set that is of b ounded tree
w
c w
width has a nite numb er of obstructions A b ound O on the maximum
numb er of edges of an obstruction is given in terms of the treewidth b ound w and
the index c of a nite congruence
Lag J Lagergren The nonexistence of reduction rules giving an emb edding into a
k tree Discrete Applied Mathematics
Lag J Lagergren Upp er b ounds on the size of obstructions and intertwines
Dept of Numerical Analysis and Computing Science Manuscript The Royal
Institute of Technology Stockholm Sweden
The authorproves constructively that if a minor closed family L has obstructions
of treewidth at most k and there is a pseudo minororder with nite height for L
then L has a nite numb er of obstructions For another pseudo minororder an
upp er b ound on the size of an intertwine of two given planargraphs H and H is
also obtained
Lan M A Langston Current progress on obstruction based layout optimiza
tion Technical rep ort Dept of Computer Science UniversityofTennessee
Knoxville TN
Lau C Lautemann Decomp osition trees Structured graph representation and ef
cient algorithms In Proceedings of the th Col loquium on Trees in Algebra
and Programming CTAP volume of Lecture Notes on Computer Sci
ence pages Springer Verlag
LB C G Lekkerkerker and J C Boland Representation of a nite graph by a set
of intervals on the real line Fundamenta Mathematicae
Len T Lengauer Black white p ebbles and graph separation Acta Informatica
Lei F Leighton New lower b ound techniques for VLSI In IEEE Symposium on
Foundations of Computer ScienceProceedings volume pages
LG P Liu and R Geldmacher An O max m n algorithm for nding a subgraph
homeomorphic to K Congressus Numerantium Also in
the Pro ceedings of the Eleventh Southeastern Conference on Combinatorics
Graph Theory and Computing
Lin A Lingas Subgraph isomorphism for biconnected outerplanar graphs in cubic
time Theoretical Computer Science
LPS G T Linger T Politof and A Satyanarayana A forbidden minor character
ization and reliability of a class of partial trees Networks
Gives seven minor order obstructions to a family of graphs that is contained in
the set of partial trees
LR M A Langston and S Ramachandramurthi Dense layouts for series parallel
circuits Technical rep ort Dept of Computer Science UniversityofTennessee
Knoxville TN
Uses K as the only test for the Gate Matrix Layout problem with three tracks
LT R J Lipton and R E Tarjan A separator theorem for planar graphs SIAM
Journal on Applied Mathematics
Lu X Lu Finite state prop erties of b ounded pathwidth graphs Master s pro ject
rep ort Dept of Computer Science University of Victoria P O Box
Victoria B C Canada V W P
Mah N V R Mahadar Master s thesis UniversityofWaterlo o
Found checked that there were minor order minimal graphs within the
previously known top ological projective plane obstructions
Man J B Manning Geometric Symmetry in Graphs Ph D thesis Purdue Univer
sity December
MCH S L Mitchell E J Co ckayne and S T Hedetniemi Linear algorithms on
recursiv e representations of trees Journal of Computer and System Sciences
The authors claim that their algorithm techniques generalize to k trees and chordal
graphs
Mil E C Milner Basic WQO and BQO theory InI Rival editor Graphs and
Orders volume of Mathematical and Physical Sciences pages D
Reidel Publishing Company Amsterdam
A survey of the achievements in the theory of well quasi wqo and b etter quasi
ordering b qo since the early s
Mir B G Mirkin Graphs and Genes Springer Verlag translated from
Russian by H L Beus
Mit S L Mitchell Linear algorithms to recognize outerplanar and maximal outer
planar graphs Information Processing Letters Decemb er
MK P J McGuinnes and A E K ezdy An algorithm to nd a K minor
An O n algorithm is presented that given a graph either returns a K minor
or rep orts that no such minor exists
MM E Mata Montero Resilience of partial k tree networks with edge and no de
failures Networks
M oh R H M ohring Graph problems releted to gate matrix layout and PLA folding
In G Tinhofer E Mayr H Noltemeier and M S in co op eration with R Al
brecht editors Computational Graph Theory Computing Supplementum
pages Springer Verlag
Graph problems o ccurring in linear VLSI layout architectures such as gate matrix
layout pathwidth folding of programmable logic arrays and Weinb erger arrays
are surveyed
Moh B Mohar Pro jective planarity in linear time Journal of Algorithms
First such linear time algorithm for testing projective planarity based on the em
b eddings of K and K
Mo o J Mo on The numb er of lab eled k trees Journal of Combinatorial Theory
Series B
MPS F Makedon C H Papadimitriou and I H Sudb orough Top ological band
width SIAM Journal on Algebraic Discrete Methods
MRS R Motwani A Raghunathan and H Saran Constructive results from graph
minors Linkless emb eddings In IEEE Symposium on Foundations of Com
puter ScienceProceedings pages th F OCS White Plains
N Y
The authors prove that there are seven linkless emb eddable minor minimal ob
structions
MS F Makedon and I H Sudb orough Minimizing width in linear layouts In
Proceedings of the International Col loquium on Automata Languages and Pro
gramming volume of Lecture Notes on Computer Science pages
Springer Verlag Also see Discrete Applied Mathematics
Contains forbidden subgraphs forcutwidth and also characterizes cutwidth
family A dynamic programming algorithm is improved to show that forany k
the problem of deciding if a given graph has cutwidth at most k can b e done in
k
O n
MS B Monien and I H Sudb orough Min cut is NP complete for edge weighted
trees Theoretical Computer Science Also in Lecture Notes
in Computer Science pp
The Min Cut Linear Arrangement Problem cutwidth is NP complete forpla
nar graphs with maximum vertex degree three This is used to show that the
Search Numb er Vertex Separation pathwidth Progressive Black White Pebble
Demand and Top ological Bandwidth is NP complete forplanar graphs with max
imum vertex degree three
MT J Matou sek and R Thomas Algorithms nding tree decomp ositions of graphs
Journal of Algorithms
A probalistic algorithm mayb e practical is given with execution time
O n log n n log nj log pj that nds either a tree decomp osition of width k
oranswers that the tree width is k
MT J Matou sek and R Thomas On the complexity of nding iso and other
morphisms for partial k trees Discrete Mathematics
Nar C Narayanan Fast paral lel algorithms and enumeration techniques for partial
k trees Ph D dissertation Dept of Computer Science Clemson University
August
Gives an overview of parallel algorithms forpartial k trees
Nar C Narayanan Isomorphism testing of k trees is in NC for xed k Information
Processing Letters
Ner A Nero de Linear automata transformations Proceedings of the American
Mathematical Society
Neu E T Neufeld Practical toroidality testing Master s thesis Dept of
Computer Science University of Victoria P O Box Victoria B C
Canada V W P
NW C S J A Nash Williams On well quasi ordering nite trees Proceedings of
the Cambridge Philosophical Society
Presents a simple pro of that any sequence of nite trees T T there exists
i j such that T is homeomorphic to a subtree of T
i j
OvW R Otten and J van Wijk Graph representations in interactivelayout design
In Proceedings of the IEEE SymposiumonCircuits and Systems pages
Par T D Parsons Pursuit evasion in a graph In Y Alavi and D R Lick editors
Theory and applications of graphs pages Springer Verlag
PLH J Pfa R Laskar and S T Hedetniemi Linear algorithms for indep endent
domination and total domination in series parallel graphs Congressus Numer
antium
Pou M Pouzet Applications of well quasi ordering and b etter quasi ordering In
I Rival editor Graphs and Orders volume of Mathematical and Physical
Sciences pages D Reidel Publishing Co
Pro A Proskurowski Minimum dominating cycles in trees International Journal
of Computation and Information Science
Pro A Proskurowski Separating subgraphs in k trees Cables and caterpillars
Discrete Mathematics
Pro A Proskurowski Graph reductions and techniques for nding minimal for
bidden minors In N Rob ertson and P D Seymour editors Graph Structure
Theory volume of Contemporary Mathematics pages American
Mathematical So ciety
A brief historical account of some techniques used to nd small sets of minimal
forbidden minors fo r a few classes of graphs with treewidth at most is given
PRS A Proskurowski F Ruskey and M Smith Listing k paths Technical Rep ort
DCS IR Dept of Computer Science University of Victoria P O Box
Victoria B C Canada V W P June
A simple and e cient algorithm for enumerating k paths represented as k ary strings is given
PS A Proskurowski and M M Syslo Minimum dominating cycles in outerpla
nar graphs International Journal of Computation and Information Science
PS A Proskurowski and M M Syslo E cient computations in tree like graphs
In G Tinhofer E Mayr H Noltemeier and M S in co op eration with R Al
brecht editors Computational Graph Theory Computing Supplementum
pages Springer Verlag
This is a go o d survey pap er on e cient algorithms forpartial k trees
PS T Politof and A Satyanarayana Minors of quasi connected graphs Discrete
Mathematics
Ram S Ramachandramurthi Algorithms for VLSI Layout BasedonGraph Width
Metrics Ph D dissertation Dept of Computer Science UniversityofTen
nessee Knoxville TN August
Ree B A Reed Finding approximate separators and computing tree width quickly
In Proceedings of the ACM Symposium on the TheoryofComputing volume
pages
Gives an approximate separatoralgorithm that yields an O n log n algorithm that
nds a width k for xed k tree decomp osition if it exists
Ric M B Richey Combinatorial optimization on series paral lel graphs algorithms
and complexity Ph D dissertation Scho ol of Industrial and Systems Engineer
ing Georgia Institute of Technology
Ros D J Rose On simple characterizations of k trees Discrete Mathematics
Contains four simple characterizations of k trees involving cliques paths and
separators
RS N Rob ertson and P D Seymour Graph Minors I Excluding a Forest Journal
of Combinatorial Theory Series B
RS a N Rob ertson and P D Seymour Graph Minors I I Algorithmic asp ects of
tree width Journal of Algorithms
RS c N Rob ertson and P D Seymour Graph Minors I I I Planar tree width Jour
nal of Combinatorial Theory Series B
RS a N Rob ertson and P D Seymour Graph Minors IV Tree width and well
quasi ordering Journal of Combinatorial Theory Series B
RS d N Rob ertson and P D Seymour Graph Minors V Excluding a planar graph
Journal of Combinatorial Theory Series B
RS b N Rob ertson and P D Seymour Graph Minors VI Disjoint paths across a
disc Journal of Combinatorial Theory Series B
RS N Rob ertson and P D Seymour Graph Minors VI I Disjoint paths on a
surface Journal of Combinatorial Theory Series B
RS c N Rob ertson and P D Seymour Graph Minors VI I I A Kuratowski theorem
for general surfaces Journal of Combinatorial Theory Series B
RS b N Rob ertson and P D Seymour Graph Minors IX Disjoint crossed paths
Journal of Combinatorial Theory Series B
RS a N Rob ertson and P D Seymour Graph Minors X Obstructions to tree
decomp ositions Journal of Combinatorial Theory Series B
RS N Rob ertson and P D Seymour Graph Minors XI Circuits on a surface
Journal of Combinatorial Theory Series B
RS a N Rob ertson and P D Seymour Graph Minors XI I Distance on a surface
Journal of Combinatorial Theory Series B July
RS b N Rob ertson and P D Seymour Graph Minors XI I I The disjointpaths
problem Journal of Combinatorial Theory Series B
RS b N Rob ertson and P D Seymour Graph Minors XIV Extending an embed
ding submitted
RSa N Rob ertson and P D Seymour Graph Minors XV Giant steps submitted
RSb N Rob ertson and P D Seymour Graph Minors XVI Excluding a non planar
graph submitted
RS N Rob ertson and P D Seymour Graph Minors XVI I Taming a vortex May
submitted
RSh N Rob ertson and P D Seymour Graph Minors XVI I I Tree decomp ositions
and well quasi ordering in progress
RSg N Rob ertson and P D Seymour Graph Minors XIX Well quasi ordering on
a surface in progress
RSc N Rob ertson and P D Seymour Graph Minors XX Wagner s conjecture in
progress
RSd N Rob ertson and P D Seymour Graph Minors XXI Graphs with unique
linkages submitted
RSe N Rob ertson and P D Seymour Graph Minors XXI I Irrelevantvertices in
linkage problems submitted
RSf N Rob ertson and P D Seymour Graph Minors XXI I I Nash Williams im
mersions conjecture in progress
RS J Riordan and C E Shannon The numberoftw o terminal series parallel
networks Journal of Mathematical Physics
RS a N Rob ertson and P D Seymour Graph width and well quasi ordering a
survey In J Bondy and U Murty editors Progress in Graph Theory pages
Academic Press Inc Toronto
RS b N Rob ertson and P D Seymour Generalizing Kuratowski s theorem Con
gressus Numerantium
RS e N Rob ertson and P D Seymour Some new results on the well quasi ordering
of graphs In M Pouzet and D Richard editors Orders Description and
Roles volume of Annals of Discrete Mathematics pages Elsevier
Science Amsterdam Pro ceedings of the Conference on Ordered Sets and
Their Applications Ch ateau de la Tourette l Arbresle July
RS a N Rob ertson and P D Seymour Disjoint paths A survey SIAM Journal
on Discrete Mathematics
RS b N Rob ertson and P D Seymour Graph Minors A survey InSurveys in
Combinatorics volume pages Cambridge University Press
Presents a survey of results concerning Wagner s conjecture and the Disjoint Con
necting Paths problem
RS d N Rob ertson and P D Seymour An Outline of a Disjoint Paths Algorithm
pages Algorithms and Combinatorics Volume
RS a N Rob ertson and P D Seymour Excluding a graph with one crossing In
N Rob ertson and P D Seymour editors Graph Structure Theory volume
of Contemporary Mathematics pages
RS b N Rob ertson and P D Seymour editors Graph Structure Theory volume
of Contemporary Mathematics American Mathematical So ciet y Pro
ceedings of a Joint Summer Research Conference on Graph Minors held June
to July at the UniversityofWashingtion Seattle
RST c N Rob ertson P D Seymour and R Thomas A survey of linkless emb eddings
In N Rob ertson and P D Seymour editors Graph StructureTheory volume
of Contemporary Mathematics pages AMS
It is shown that a graph has a at linkless emb edding if and only if it has no
minor isomorphic to one of seven Peterson family obstructions These graphs are
obtained from K by means of Y and Y exchanges
RST b N Rob ertson P D Seymour and R Thomas Structural descriptions of lower
ideals of trees In N Rob ertson and P D Seymour editors Graph Structure
Theory volume of Contemporary Mathematics pages
RST a N Rob ertson P D Seymour and R Thomas Linkless emb eddings of graphs
in space Bul letin of the American Mathematical Society
RST N Rob ertson P D Seymour and R Thomas Quickly excluding a planar
graph Journal of Combinatorial Theory Series B
RST a N Rob ertson P D Seymour and R Thomas Kuratowski chains Journal of
Combinatorial Theory Series B
RST b N Rob ertson P D Seymour and R Thomas Petersen family minors Journal
of Combinatorial Theory Series B
RST c N Rob ertson P D Seymour and R Thomas Linkless emb edding conjecture
Journal of Combinatorial Theory Series B
RTL D J Rose R E Tarjan and G S Leuker Algorithmic asp ects of vertex
elimination on graphs SIAM Journal on Computing
Relates graph elimination to p erforming Gaussian elimination on sparse symmetric
p ositive de nite matrices
San D P Sanders On linear recognition of tree width at most four Georgia
Tech manuscript submitted
Sar H Saran Constructive Results from Graph Minors Linkless Embeddings
Ph D Thesis University of California Berkeley
Sch P Sche er An O n log n algorithm for the pathwidth of trees In Funda
mentals of Computation Theory FCT Karl WeierstraB Institut Fur
Mathematik Akademie der Wissenshaften der DDR Mohrenstr Berlin
DDR
Sch P Sche er A linear algorithm for the pathwidth of trees In R Bo dendiek
and R Henn editors Topics in combinatorics and graph theory pages
Physica Verlag
SD D C Schmidt and L E Dru el A fast backtracking algorithm Journal of
the Association Computing Machinery July
Contains a nice graph isomorphism algorithm which is very practical Weuse
this
Sey a P D Seymour A forbidden minor characterization of matroid p orts Quarterly
Journal of Mathematics Oxford
Sey b P D Seymour The forbidden minors of binary clutters Journal of London
Mathematical Society
Sey P D Seymour Colouring series parallel graphs Combinatorica
Sey P D Seymour A b ound on the excluded minors for a surface February
Bell Communications Research Morristown New Jersey
For every surface of complexity g e g twice the genus every excluded minor
k
has at most vertices where k g The treewidth is also b ounded by
g
SI M Syslo and M Iri E cient outerplanarity testing Fundamenta Informaticae
ST A Satyanarayana and L Tung A characterization of partial trees Networks
ST P D Seymour and R Thomas Graph searching and a min max theorem for
treewidth Journal of Combinatorial Theory Series B
Ste M Steinby On generalizations of the Myhill Nero de Theorems Bul letin of the
EATCS
Sti J Stillwell Classical Topology and Combinatorial Group Theory volume of
exts in Mathematics Springer Verlag second edition Graduate T
SW G Smith and R Walford The identi cation of a minimal feedbackvertex of a
directed graph IEEE Transactions on Circuits and Systems CAS
Sys M M Syslo The subgraph isomorphism problem for outerplanar graphs
Theoretical Computer Science
Sys M M Syslo NP complete problems on some tree structured graphs a review
In Proceedings of the Workshop on Graph Theoretic Concepts in Computer
Science WG pages Trauner Verlag Linz June
TGS P J Tanenbaum M T Go o drich and E R Scheinerman Characteriza
tion and recognition of p oint halfspace and related orders In Graph Drawing
DIMACS International Workshop GD Princeton N J October Pro
ceedings volume of Lecture Notes on Computer Science pages
Springer Verlag
The authors characterize four classes of geometric memb ership and containment
orders structurally in terms of forbidden subp osets
Tha J W Thatcher Tree automata an informal survey In A V Aho editor
Currents in the Theory of Computing pages Prentice Hall
This is a go o d intro duction to tree automata
Tho C Thomassen Handbook of Combinatorics chapter Em b eddings and minors
North Holland to app ear
Tho R Thomas A Menger like prop erty of tree width the nite case Journal of
Combinatorial Theory Series B
Shows the existence of linked vertex disjoint paths between bags tree
decomp ositions
Tho R Thomas private communication Dept of Mathematics Georgia
Institute of Technology
TM W T Trotter and J I Mo ore Characterization problems for graphs partially
ordered sets lattices and families of sets Discrete Mathematics
TNS K Takamizawa T Nishizeki and N Saito Linear time computability of com
binatorial problems on series parallel graphs Journal of the Association Com
puting Machinery
Shows that NP complete problems characterizable in terms of a nite numb er of
forbidden subgraphs admit linear time algorithms if their instances are restricted
to series parallel graphs
Tro W T Trotter Combinatorics and Partial ly Ordered Sets Dimension The
ory Johns Hopkins Univ Press Baltimore Johns Hopkins Series in
Mathematical Sciences
Tru R J Trudeau Introduction to Graph Theory Dover Revised edition of
Dots and Lines
TUK A Takahashi S Ueno and Y Ka jitani Minimal acyclic forbidden minors
for the family of graphs with b ounded path width In SIGAL IPSJ
To app ear in Annals of Discrete Mathematics Pro ceedings of nd Japan
conference on graph theory and combinatorics
Shows that there are at least k tree obstructions for the family of graphs with
pathwidth at most k
TW J W Thatcher and J B Wright Generalized nite automata Notices of the
American Mathematical Society
TW J W Thatcher and J B Wright Generalized nite automata theory with an
application to a decision problem of second order logic Mathematical System
Theory
Many of the imp ortant concepts and results of conventional nite automata theory
are develop ed for a generalization in which nite algebras take the place of nite
automata
vL J van Leeuwen Handbook of Theoretical Computer Science A Algorithms
and Complexity Theory MIT Press North Holland Publishing Company
Amsterdam See Graph Algorithms chapter in pages
This handb o ok contains several sections regarding combinatorial b ounded width
and forbidden minors
Vol W Vollmerhaus On computing all minimal graphs that are not emb eddable
in the pro jective plane parts I and I I cited manuscript
The author indep endent ly veri ed the projective plane irreducible graphs and
may have also veri ed the mino r minimal graphs
VTL J Valdes R Tarjan and E Lawler The recognition of series parallel digraphs
SIAM Journal on Computing Also see STOC pages
Presents a vertex forbidden subgraph for vertex series parallel DAGs in the
transitive closure
Wag K Wagner Ub er einer eigenschaft der eb ener complexe Mathematics Annals
WC J A Wald and C J Colb ourn Steiner trees partial trees and minimum IFI
networks Networks
Studying isolated failure immune networks the authors nd a linear time Steiner
tree algorithm for trees
n
Weg G Wegner Eigenshaften der nerven homologische eihfacter familien in R
Ph d thesis G otingen
Gives three induced subgraph obstructions asteroidal triples for chordal graphs
which are not prop er interval graphs
WHL T V Wimer S T Hedetniemi and R Laskar A metho dology for constructing
linear time graph algorithms Congressus Numerantium
Wil R J Wilson Introduction to Graph Theory Academic Press Inc
Wil H S Wilf Finite list of obstructions the editor s corner American Mathe
matics Monthly pages March
An intro duction to the notion of obstructions and intro duces the work of Rob ert
son and Seymour
Wil H S Wilf Combinatorial Algorithms AnUpdate volume of CBMS NSF
Regional Conference Series in Applied Mathematics chapter Listing Free Trees
pages SIAM
Wim a T V Wimer Linear algorithms for the dominating cycle problems in series
parallel graphs trees and Halin graphs Congressus Numerantium
Wim b T V Wimer Linear algorithms on k terminal graphs Ph D dissertation
Dept of Computer Science Clemson University August Rep ort No
URI
Win P Winter Generalized Steiner problem in series parallel networks Journal of
Algorithms
A linear time algorithm is given for a generalized Steiner problem
WROM R A Wright B Richard A Odlyzko and B D McKay Constant time
generation of free trees SIAM Journal on Computing
WWY J Wang D B West and B Yao Maximum bandwidth under edge addition
Journal of Graph Theory
Yap C K Yap Some consequences of non uniform conditions on uniform classes
Theoretical Computer Scienc e
The main result shows that if the non uniform p olynomial time hierarchy col
lapses at level i i e p oly p oly then the Meyer Sto ckmeyer hierarchy
i i
P
collapses at level i i e
i
i
ZiNN X Zhou S ichi Nakano and T Nishizeki A parallel algorithm for edge
coloring partial k trees In Proceedings of the rdScandinavian Workshop on
Algorithm Theory volume of Lecture Notes on Computer Science pages
Springer Verlag
Index
k Vertex Cover asteroidal triples
K cover automata
minimal approximating
outer planar maximum path
minimum testset problem AMT
graph
tree
connected
edge contraction bandwidth
edge lift biology
edge sub division b oundaried graphs
minimal b oundaried obstruction
order b oundary minor order
p ebbling broadcast graphs
size
C
vertex fracture
cactus
graph isomorphism
canonical congruence
Graph Minor Theorem GMT
Halin graphs second order
chordal graphs Hamiltonian cycle
circle plus op erator Hamiltonian path
clique hyp eredge cover
IBM
communication networks
immersion order
Cray Y MP
induced
cutwidth
indep endent set
cycle cover
induced subgraph
xed
integer partitions
disconnected pruning interval graphs
distributive programming colored ICG
dynamic programming
k Cycle Cover l
Eulor s surface formula k Clique and k Clique
k Clique Cover c
feedback edge set FES
k Cutwidth
feedbackvertex set FVS
k Edge Bounded IndSet
nite index congruence
k Fixed Cycle Cover c
k Edge Bounded IndSet
k Feedback Edge Set
path cover
k Feedback Vertex Set
nite state algorithm
k Edge Bounded IndSet
k Genus
k Feedback Vertex Set
k IndSet and k IndSet
Outer Planar quasi order
k Path Cover p
relational databases
Planar
k Pathwidth
second order monadic logic
k Treewidth
series parallel graphs
k Vertex Cover
Sparc
K cover
knottless graphs
TCP IP
Kuratowski s Theorem
testset
k Feedback Edge Set
LEDA library
k Feedback Vertex Set
lex canonic order
connected graph
lex rank order
cycle cover
linear algebra
Hamiltonian
linguistics
maximum path cycle
linkless graphs
top ological order
Los Alamos National Lab oratory
toroidal graphs
lower ideal
t parse
parameterized
canonic
concatenation
minimum test collection MTC
distinguisher
minor order
enumeration
edge colored
extension
induced
minimal
minor containment problem
pre x
Myhill Nero de Theorem
treewidth canonic
nonminimal pretests t parse isomorphism
k Feedback Edge Set free xed b oundary
NP complete tree
tree decomp osition
obstruction set
smo oth
partial
treewidth
op erator sets
treewidth search
outer planar graphs
unit interval problem
partial t paths
universal distinguisher
partial t trees
University of Victoria
partial order
UNIX shell scripts
path cover
path decomp osition VACS
smo oth vertex cover VC
pathwidth vertex separation
colored CVS
closeness to width t virtual b oundary
Petersen graph VLSI
planar graphs
weak immersion order
p olynomial time hierarchy
well partial order
p oset
well quasi order
Pre x Lemma
pro jective plane
Vita
Surname Dinneen Given Names Michael John
Michael was b orn in Idaho Falls Idaho U S A and lived there with his parents
two brothers and a sister until he left for undergraduate scho ol After receiving two
bachelor degrees he enrolled in graduate scho ol in Michael s rst real job was
a computer programming job at Idaho Falls Scho ol District for just over four
years during the scho ol breaks of his undergraduate career Recently Michael has
b een working at Los Alamos National Lab oratory New Mexico as a Graduate Re
search Assistant for the Computer Research and Applications Group for an average
of six months a year since
Educational Institutions Attended
University of Victoria Victoria B C
Washington State University Pullman WA
University of Idaho MoscowID
Degrees Awarded
M S Computer Science University of Victoria May
Thesis Algebraic Metho ds for E cientNetwork Constructions
B S Computer Science Magna Cum Laude UniversityofIdaho May
B S Mathematics Applied Cum Laude UniversityofIdaho May
Awards and Scholarships
University of Victoria Teaching Fellowship
University of Victoria Fellowship
Washington State Universit yTeaching Assistantship
Chevron Oil s Science and Engineering Scholarship
Taylor Eugene and Osa Scholarship
J Lawrence Botsford Scholarship
General University of Idaho Scholarship
University of Idaho Dean s List College of Engineering
Faculty of Letters and Science
Selected Pap ers
Algebraic constructions of e cient broadcast networks In H F Mattson
T Mora and T R N Rao editors AppliedAlgebra Algebraic Algorithms
and Error Correcting Codes volume of Lecture Notes in Computer Science
pages Springer Verlag Octob er Pro ceedings of the th Interna
tional Symp osium AAECC with Vance Fab er and Michael R Fellows
Small diameter symmetric networks from linear groups IEEE Transactions
on Computers with Lowell Campb ell Gunnar E Carlsson
Vance Fab er Michael R Fellows Michael A Langston James W Mo ore An
drew P Mullhaupt and Harlan B Sexton
Recent examples in the theory of partition graphs Discrete Mathematics
with Duane W DeTemple Kevin L McAvaney and JackM
Rob ertson
A characterization of graphs with vertex cover up to ve In V Bouchitte and
M Morvan editors Orders Algorithms and Applications ORDAL volume
of Lecture Notes in Computer Science pages Springer Verlag July
with Kevin Cattell
New results for the degree diameter problem Networks Octob er
with Paul R Hafner
Obstructions to within a few vertices or edges of acyclic In Pro ceedingsofthe
Fourth Workshop on Algorithms and Data Structures WADS volume
of Lecture Notes in Computer Science pages Springer Verlag August
with Kevin Cattell and Michael R Fellows
A computational attack on the conjectures of Gra ti New counterexamples
and pro ofs Discrete Mathematics In press accepted with TonyL
Brewster and Vance Fab er
A simple linear time algorithm for nding path decomp ositions of small width
Accepted Information Processing Letters with Kevin Cattell and Michael R
Fellows