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3.2 the Graph Minor Theorem Bounded Combinatorial Width and Forbidden Substructures by Michael John Diimeen B.S., University of Idaho, 1989 M.S., University of Victoria, 1992 A Dissertation Submitted in Partial Fulfil'ment 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, Supervisor (Department of Computer Science) Dr. Hausi A. Muller, Department Member (Department of Computer Science) Dr. Jon C. ^MuzibfDepartment Member (Department of Computer Science) Dr. Garf^MacGilliv^, OutsideWmber (Department of Mathematics) ^ / ■ — —-• Dr. Arvind Gupta, External Examiner (School of Computing Science, Simon Fraser University) © MICHAEL JOHN DINNEEN, 1995 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by mimeograph or other means, without the permission of the author. ii Supervisor: M. R. Fellows A bstract A substantial part of the history of graph theory deals with the study and classifi­ cation of sets of graphs that share common properties. 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 classifies the planar graph family by two forbidden graphs (in the topological partial order). Most, if not all, of the current approaches for finding these forbidden substructure characterizations use extensive and specialized case analysis. Thus, until now, for a fixed graph family, this type of mathematical theorem proving often required months or even years of human effort. The main focus 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 importantly) implement a variation of the seminal work done by Fellows and Langston regarding computing finite-basis characterizations, The recently celebrated Robertson-Seymour Graph Minor Theorem establishes that many natural graph families are characterizable by a finite set of graphs. In particular, if a graph family is closed under the three basic minor operations (i.e., isolated vertex deletions, edge deletions, and edge contractions) then there exists (by a nonconstruc­ tive argument) a finite set of forbidden graphs. Two examples are the well-known /..•-V ertex C over , and ^-F eedback V ertex S et graph families. In this disserta­ tion, we characterize, for the first time, these parameterized families, among others, for small /,:. Our forbidden graph computations use a restricted search space consisting of graphs of bounded combinatorial wi ,Ji (where pathwidth and treewidth are two important metrics). Using an algebraic enumeration scheme for graphs, we have implemented a terminating algorithm that will find all minor-order forbidden graphs for each fixed pathwidth. For a targeted graph family, this algorithm requires a mathematical description given in one of many acceptable forms (or combinations thereof), such as a finite-index congruence or a set of automaton-generating tests. Our main assumption is that an upper bound on the pathwidth (or treewidth) of the largest forbidden graph of a particular graph family is more readily available than its order (or size). A byproduct of our bounded width approach is that we give practical linear time membership algorithms in the form of dynamic programs (over parsed graph struc­ tures of bounded width) for several graph families (e.g., /c -M aximum P ath L eng th and O u t e r P lanar ). >11 Examiners: Dr. Michael R. Fellows, Supervisor (Department of Computer Science) Dr. Hausi A. Muller, Department MembeX' (Department of Computer Science) Dr. Jon C. Muzi^f^ephrtment Member (Department of Computer Science) Dr. G ary^lacG illivr^, OutsideCE^ember (Department of Mathematics) Dr. Arvind Gupta, External Examiner (School of Computing Science, Simon Fraser University) iv Contents Abstract ii Table of Contents iv List of Figures viii List of Tables xi Acknowledgements xii 1 Introduction 1 1.1 The Graph Theory Setting ......................................................................... 5 1.2 Some Finitely Characterizable Graph Families ....................................... 9 1.3 A Historical Perspective on Computing O bstructions .......................... 12 1.4 An Overview of Our Computational Technique ....................................... 13 1.5 A Survey of The Dissertation ................................................................... 18 1 The Basic Theory 21 2 Bounded Combinatorial Width 22 2.1 Introduction ................................................................................................... 23 2.1.1 Treewidth and k-trees .................................................................... 24 2.1.2 Pathwidth and k-p a th s .................................................................... 26 2.1.3 Other graph-theoretical widths .................................................... 31 2.2 Algebraic Graph Representations ............................................................. 33 V 2.2.1 The 4-parse operator s e t ................................................................ 34 2.2.2 Some 4-parse exam ples ................................................................... 40 2.2.3 Other operator s e ts .......................................................................... 41 2.3 Simple Enumeration S c h e m es .................................................................... 45 2.3.1 Canonic pathwidth 4 -p a rse s .......................................................... 46 2.3.2 Canonic treewidth 4-parses ............................................................. 51 3 Graph Minors and Well-Quasi-Orders 56 3.1 Preliminaries . ......................................................................................... 56 3.2 The Graph Minor Theorem ...................................................................... 59 3.3 Other Graph Partial O rd e rs ...................................................................... 65 4 Finding Forbidden Minors 68 4.1 Key 4-parse P ro p e rtie s ............................................................................... 68 4.2 Simple Procedure for Finding O b stru ctio n s ...................................... 72 4.3 Proving 4-parses Minimal or Nonm inim al ................................................ 75 4.3.1 Direct proofs of nonminimalifcy ................................................... 76 4.3.2 Proofs based on a dynamic programming algorithm ................. 77 4.3.3 Proofs obtained by a randomized search .................................... 83 4.3.4 Proofs based on a testset congruence .......................................... 85 4.4 Making the Theory Practical ...................................................................... 87 4.4.1 Pruning at disconnected 4-parses ................... 87 4.4.2 Searching via universal distinguishes .......................................... 91 4.4.3 Using other linite-index congruences .......................................... 93 4/1.4 Finding uses of random ization ....................................................... 94 5 The Implementation and The Future 96 5.1 Using Distributed Programming ................................................................ 96 5.2 Software S u m m ary ...................................................................................... 98 5.3 Future Research G oals ................................................................................ 105 5.3.1 Second-order congruence research ................... 107 5.3.2 Approximation algorithms based on partial obstruction sets . 109 vi II Obstruction Set Characterizations 112 6 Vertex Cover (VC) 113 6.1 Introduction .................................................................................................. 113 6.2 A Finite State Algorithm ........................................................................... 115 6.3 The VC Obstruction Set Computation .................................................. 121 6.4 The VC O bstructions .................................................................................. 124 6.5 Independent Set and Clique Fam ilies ..................................................... 125 7 Feedback Vertex/Edge Sets (FVS and FES) 131 7.1 Introduction ................................................................................................. 131 7.2 The FVS Obstruction Set Computation ....................................... 133 7.2.1 A finite state algorithm .................................................................... 133 7.2.2 A complete FVS testset ................................................................ 142 7.3 The FVS Obstructions ................................................................... 145 7.4 The FES Obstruction Set Computation ....................................... 149 7.4.1 A direct nonminimal FES t e s t ....................................................... 149 7.4.2 A complete FES testset ................................................................ 151 7.5 The FES Obstructions ......... 152 8 Some Generalized VC and FVS Graph Families 158 8.1 Path Covers .............................................. 158 8.1.1 A patb-cover congruence ......................... 160 8.2 Cycle Covers .............................................................................................. 164 8.3 Path/Cycle Cover T estsets .......................................................................
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