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Tmversity Microfilms International TMversity Microfilms International 1.0 lit |2jS |£ 5 U I B2.2 IS m “ s u. y o 1.1 DZ 1.25 m u MICROCOPY RESOLUTION TEST CHART NATIONAL BUREAU OF STANDARDS 4 STANDARD REFERENCE MATERIAL 1010a ! fi 5 I ANSI And ISO TEST CHART No. ?) 1):! University Mcrofilms Inc. 300 N. Zeeb Roac, Ann Arbor, Ml 48106 I ! INFORMATION TO USERS This reproduction was made from a copy of a manuscript sent to us for publication and microfilming. While the most advanced technology has been used to pho­ tograph and reproduce this manuscript, the quality of the reproduction Is heavily dependent upon the quality of the material submitted. Pages In any manuscript may have Indistinct print. In all cases the best available copy has been Aimed. The following explanation of techniques Is provided to help clarify notations which may appear on this reproduction. 1. Manuscripts may not always be complete. 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Fbr an additional charge, all photographs are available in black and white standard 35mm slide format.* •For more Information about black and white slides or enlarged paper reproductions, please contact the Dissertations Customer Services Department T T-\yf-T Dissertation L J 1 V 1 1 Information Service University Microfilms International A Belt & Howell Information Company 300 N. Zeeb Road, Ann Arbor, Michigan 48106 6625315 Weaver, Robert Wooddell SOME PROBLEMS IN STRUCTURAL GRAPH THEORY The Ohio State University Ph.D. 1986 University Microfilms International300 N. Zeeb Road, Ann Arbor. Ml 40100 PLEASE NOTE: In all cases this material has been filmed In the best possible way from the available copy. Problems encountered with this document have been identified herewith a checkmark «/ 1. Qlossy photographs or pages _____ 2. Colored illustrations, paper or print ______ 3. Photographs with dark background _____ 4. Illustrations are poor copy ______ 5. 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Other _______________________________________________________________________ University Microfilms International SOME PROBLEMS IN STRUCTURAL GRAPH THEORY DISSERTATION Presented in Partial Fulfillment of the Requirements for the Degree Doctor of Philosophy in the Graduate School of the Ohio State University By Robert Wooddell Weaver, B. S., M. Sci. * * * * * The Ohio State University 1986 Dissertation Committee: Approved by N. Robertson T. Dowling {\SU-X Adviser P. Huneke Department of Mathematics D. Ray-Chaudhuri To the memory of Herbert John Ryser ACKNOWLEDGEMENTS I express my sincerest appreciation to Dr. Neil Robertson, to whom I owe whatever knowledge of structural graph theory I possess. Thanks for his time and patience during the many years of study. Thanks also go to Dr. Thomas Zaslavsky and Joseph Fiedler, for helpful conversations that led to the development of chapters two and five. A special thanks to Dr. Paul Seymour, for his work that led to chapter four and for some exciting courses of study. To my dissertation committee, I offer gratitude for spending several hours of their summer reading this dissertation. Finally I would like to express my appreciation for Richard Vitray, fellow student of graph theory, co-conspirator, and friend, who made the time at OSU a little more pleasant and successful. VITA August 19t 1958 ................................................. Bom — San Diego, California 1980 .................................................................. B. S., California Institute of Technology, Pasadena, CA 1982 .................................................................. M. Sci., The Ohio State University, Columbus, OH PUBLICATIONS Seymour, P. D., and Weaver, R. W., A generalization of chordal graphs, J. Graph Theory, Vol. 8 , No. 2, Summer 1984. FIELDS OF STUDY Major Field: Structural Graph Theory Studies in: Matroid Theory with Paul Seymour TABLE OF CONTENTS DEDICATION................................................................................ ii ACKNOWLEDGEMENTS............................................................ Hi VITA.............................................................................................. iv LIST OF FIGURES........................................................................ vii LIST OF PLATES.......................................................................... ix LIST OF NUMBERED ITEMS...................................................... x CHAPTER PAGE I. INTRODUCTION......................................................... 1 H. HEXAGONS AND NOWHERE ZERO FIVE FLOWS . 13 Si. The main theorem ......................................... 13 §2. Ornaments and the isthmus tree. 17 53. Sets that prune trees ..................................... 2 2 54. Establishing the main theorem .................... 31 55. Applications ......................................... 38 ID. GIRTH 7 GRAPHS AND NOWHERE ZERO FIVE FLOWS. 46 51. The minimal counterexample .................... 46 52. The case v=24 ............................................. 60 53. The case v«26............................................. 62 54. The case v=28 ........................................... 90 55. A corollary .................................................. 108 v IV. A GENERALIZATION OF CHORDAL GRAPHS. I l l 51.Histoiy of the problem. ............................... I l l 52. Contracted strangulations ............................. 116 53. Small cutsets............................................... 118 54. The main proof. ........................................... 124 V. A FLOW POLYNOMIAL IDENTITY FOR VERTEX 3-SEPARATED GRAPHS............................................. 129 51. The identity ................................................. 129 APPENDIX PASCAL PROGRAMS USED IN CHAPTER H I 137 Results of the v«28 automated search ............................. 188 LIST OF REFERENCES................................................................... 211 vi LIST OF FIGURES 1 Some elementaiy definitions .................................................. 2 2 Some special graphs ............................................................... 3 3 How contractions are obtained............................................... 4 4 A typical 3-separation ............................................................ 5 5 An isthmus .............................................................................. 6 6 A Z^flow on a 4-clique ................ 8 7 Converting from <pq to <pj ...................................................... 16 8 The two trees with 4 monovalent vertices ........................... 25 9 The three trees with five monovalent vertices ..................... 27 10 The seven trees with six monovalent vertices ....................... 29 11 Edges that must be avoided .................................................... 29 12 A binary branching tree .......................................................... 49 13 An infinite family .................................................................. 50 14 Some sporadic examples ........................................................ 50 15 Three of the T parts ................................................................ 53 16 Two possibilities .................................................................... 55 17 The graph G j .......................................................................... 56 18 The graph G 2 .......................................................................... 6 6 19 Two types of crossing pairs .................................................... 57 20 The graph G 3 ......................................................................... 58 21 The graph G 4 ............................. 58 22 The 7-cage, the McGee graph ................................................ 60 23 D2 with vertex labeling .......................................................... 62 24 T2 with vertex labeling .............................'.... .................... 63 25 T3 with vertex labeling ......................................... 64 26 Two indecomposable girth 5 graphs with 12 vertices 65 27 Sj with ab, cd, a l, b3, c 2 , d4,67 subdivided .............. 6 8 28 Beth ....................................................................................... 70 29 C asper ................................................................................. 71 30 Si with
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