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Bounded Combinatorial Width and Forbidden Substructures Bounded Combinatorial Width and Forbidden Substructures by Michael John Dinneen BS University of Idaho MS University of Victoria A Dissertation Submitted in Partial Fulllment 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 classies 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 familythistyp e of mathematical theorem proving often required months or even years of human eort 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 theoryWe extend and more imp ortantly implementa variation of the seminal work done byFellows and Langston regarding computing nitebasis characterizations The recently celebrated Rob ertsonSeymour 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 ie 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 wellknown 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 minororder forbidden graphs for eachxed pathwidth For a targeted graph family this algorithm requires a mathematical description given in one of many acceptable forms or combinations thereof suchasa niteindex congruence or a set of automatongenerating 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 eg 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 graphtheoretical widths Algebraic Graph Representations v The tparse op erator set Some tparse examples Other op erator sets Simple Enumeration Schemes Canonic pathwidth tparse s Canonic treewidth tparses Graph Minors and WellQuasiOrders Preliminaries The Graph Minor Theorem Other Graph Partial Orders Finding Forbidden Minors Key tparse Prop erties A Simple Pro cedure for Finding Obstructions Proving tparses 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 tparses Searching via universal distinguishers Using other niteindex congruences Finding uses of randomization The Implementation and The Future Using Distributed Programming Software Summary Future Research Goals Secondorder 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 FeedbackVertexEdge 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 pathcover congruence Cycle Covers PathCycle Cover Testsets Some maximum pathcycle testsets A maximum path automaton example Some generic cyclecover testsets Other VCFVS Generalizations The Path and Cycle Cover Obstructions OuterPlanar Graphs Intro duction vii The Outer Planar Computation An outerplanar congruence A nitestate 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 NPhard 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 LinearTime Pathwidth Algorithm Further Directions Conclusion A Summary of the Main Results Two Key Applications Annotated Bibliography Index viii List of Figures Kuratowskis planar obstructions Illustrating a partial order of graphs and a familys obstructions O F Pro jective plane emb eddings of Kuratowskis planar obstructions Induced forbidden chordal graphs asteroidal triples Demonstrating the edge contraction op eration for the minor order All connected minororder obstructions for k Edge Bounded IndSet for k Some nontrivial knots The minororder 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 familys obstructions pathwidth Demonstrating graphs with b ounded treewidth and pathwidth A partial order of several b oundedwidth families of graphs The obstructions to a pathwidth and b treewidth The minororder obstructions to treewidth Illustrating an edge contraction p oset A map from b oundaried graphs to edgecolored graphs Illustrating the a lift and b fracture graph op erations The weak immersion order obstruction set for Cutwidth Atypical tparse search tree each edge denotes one op erator ix A general indep endence algorithm I dp for pathwidth tparses The set T of b oundaried graph tests for Hamiltonicity HC Schematic view of our distributed obstruction set software Some available runtime options for obstruction set computations Building an NFA that accepts the union of minorcontainment families A general vertex cover algorithm for tparses 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 tparses Connected obstructions for Feedback
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