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Geodetic Graphs and Convexity. Michael Franklyn Bridgland Louisiana State University and Agricultural & Mechanical College

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Geodetic Graphs and Convexity. Michael Franklyn Bridgland Louisiana State University and Agricultural & Mechanical College Louisiana State University LSU Digital Commons LSU Historical Dissertations and Theses Graduate School 1983 Geodetic Graphs and Convexity. Michael Franklyn Bridgland Louisiana State University and Agricultural & Mechanical College Follow this and additional works at: https://digitalcommons.lsu.edu/gradschool_disstheses Recommended Citation Bridgland, Michael Franklyn, "Geodetic Graphs and Convexity." (1983). LSU Historical Dissertations and Theses. 3917. https://digitalcommons.lsu.edu/gradschool_disstheses/3917 This Dissertation is brought to you for free and open access by the Graduate School at LSU Digital Commons. It has been accepted for inclusion in LSU Historical Dissertations and Theses by an authorized administrator of LSU Digital Commons. For more information, please contact [email protected]. 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Uni International 300 N. Zeeb Road Ann Arbor, Ml 48106 8409573 Bridgland, Michael Franklyn GEODETIC GRAPHS AND CONVEXITY The Louisiana State University and Agricultural and Mechanical Ph.D. Col. 1983 University Microfilms International300 N. Zeeb Road, Ann Arbor, Ml 48106 PLEASE NOTE: In all cases this material has been filmed in the best possible way from the available copy. Problems encountered with this docum ent have been identified here with a check mark V . 1. Glossy photographs or pages______ 2. Colored illustrations, paper or print______ 3. Photographs with dark background______ 4. Illustrations are poor cop_______ y 5. Pages with black marks, not original copy_______ 6. Print shows through as there is text on both sid es of page_______ 7. Indistinct, broken or small print on several p ages _ 8. Print exceeds margin requirements______ 9. Tightly bound copy with print lost in spine_______ 10. Computer printout pages with indistinct print______ 11. P age(s)____________ lacking when material received, and not available from school or author. 12. P age(s)_____________seem to be missing in numbering only as text follows. 13. Two pages num bered_____________ . Text follows. 14. Curling and wrinkled p a g es_______ 15. Other_____________________________________ _____________________________________ University Microfilms International GEODETIC GRAPHS AND CONVEXITY A Dissertation Submitted to the Graduate Faculty of the Louisiana State University and Agricultural and Mechanical College in partial fulfillment of the requirements for the degree of Doctor of Philosophy i n The Department of Mathematics by Michael Franklyn Bridgland -S., Florida Technological University, 1977 M.S., Louisiana State University, 1979 December 1983 Acknowledgements In preparing this dissertation, I have enjoyed the bene-fit of much valuable advice and assistance. 1 am parti­ cularly grateful to my major professor, Dr. K. B. Reid, for his meticulous criticism of this work, which resulted in substantial improvements; for his interest in and contribu­ tions to my mathematical development; and for his seemingly boundless patience and good humor. I am also grateful to the other members of my examination committee — Dr. D. Buell, Dr. H. S. Butts, Dr. J. R. Dorroh, Dr. R. J. Koch, Dr. R. L. Lax, and Dr. J. B. Oxley — for furthering my mathematical education in numerous ways, and for providing a pleasant atmosphere of collegiality. I am especially indebted to Professor Oxley for suggesting several improve­ ments and corrections to this dissertation, and to Professor Butts for his active interest in my mathematical training, and for many stimulating discussions. Although it would be impossible to name all of the many others who have aided me in various ways during the develop­ ment of this dissertation, I certainly want to express my appreciation to Dr. B. Bollobcts, Dr. R. E. Jamison, and Dr. A. Wells for their interest and encouragement, and for the mathematical insights that they have shared with me. Finally, I thank, my wife Donna for her patience, en­ couragement, and support, without which this undertaking would have been immeasurably more difficult. i i Table of Contents Acknowledgements............................................ ii List o-f Tables.............................................. iv List of Figures........................................... v Abstract..................................... vi I. Notation, Terminology, and Historical Background. 1 II. Convexity in Graphs.................................. 20 III. Uniqueness and Ex tendi bi 1 i ty of Paths.............35 IV. Metric Structure in Geodetic Graphs........ 51 V. The Structure of Ultrageadetic Graphs............... 68 VI. A Classification of U1trageodetic Blocks. ..... 94 Bibliography............ 142 Vita......................................................... 146 i i i List o-f Tables Some generating -families for path— al ignments........... 24 1 v List of Figures 1. A geodetic graph with an induced even cycle. 8 2. Two forbidden subgraphs for geodetic graphs. 8 3. A geodetic graph of diameter three..................12 4. Four views of the Petersen graph..................... 15 5. A hierarchy of generating families. ................. 25 6 . A hierarchy of path—al ignments.......................... 25 7. A hierarchy of implications..............................30 8 . A geodetic graph of diameter four....................... 40 9. Two canonical cycles in a double chordless path. 44 10. Construction of a pyramid................... 60 11. Suspension of a graph................................ 65 12. A geodetic graph suspended from a vertex...........65 13. Suspension of an ultrageodetic graph from a clique. 76 14. Suspension of an ultrageodetic graph from a vertex. 76 15. A geodetic graph of diameter two........................ 95 16. An ul trageodet i c graph of diameter four............... 125 17. An ul trageodeti c graph of diameter four............... 138 v Abstract A graph is geodetic if each two vertices are joined by a unique shortest path. The problem of characterizing such graphs was posed by Ore in 1962; although the geodetic graphs of diameter two have been described and classified by Stemple and Kantor, little is known of the structure of geo­ detic graphs in general. In this work, geodetic graphs are studied in the context of convexity in graphs: for a suit­ able family IT of paths in a graph G, an induced subgraph H of G is defined to be IT—convex if the vertex—set of H includes all vertices of G lying on paths in IT joining two vertices of H. Then G is TT-geodetic if each TT-con— vex hull of two vertices is a path. For the family T of geodesics (shortest paths) in G, the T—geodetic graphs are exactly the geodetic graphs of the original definition. For various families H , the IT-geodetic graphs are character­ ized. The central results concern the family T of chord- less paths of length no greater than the diameter; the T~ geodetic graphs are called ultrageodetic. For graphs of di­ ameter one or two, the ultrageodetic graphs are exactly the geodetic graphs. A geometry (P,L,F) consists of an arbi­ trary set F', an arbitrary set L, and a set F £ P x L. The point—flag graph of a geometry is defined here to be the graph with vertex-set P U F whose edges are the pairs fp,<p,l)3 and { (p , 1 ) , (q , 1)3 with p,q e P, 1 e L, and vi <p,l),(q,l) e F. With the aid o-f the Feit—Higman theorem on the nonexistence of generalized polygons and the collected results o-f Fuglister, Damerel 1 -Georgi acodi s, and Damerel 1 on the nonexistence of Moore geometries, it is shown that two- connected ultrageodetic graphs of diameter greater than two are precisely the graphs obtained via the subdivision, with a constant number of new vertices, either of all of the edges incident with a single vertex in a complete graph, or of all edges of the form -Cp,(p,l)> in the point-flag graph of a finite projective plane. vi i Chapter I. Notation, Terminology, and Historical Background This report is couched in the language of graph theory; for terminology not defined here, the reader is referred to Bollob£s C53. We consider only finite simple graphs G = (V,E); that is, the vertex-set V = V(G) is a finite set, and the edge-set E = E(G) is a subset of the collec­ tion <2 > of 2-subsets of V.
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