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Maximal and Maximum Independent Sets in Graphs Western Michigan University ScholarWorks at WMU Dissertations Graduate College 6-1992 Maximal and Maximum Independent Sets in Graphs Jiuqiang Liu Western Michigan University Follow this and additional works at: https://scholarworks.wmich.edu/dissertations Part of the Applied Mathematics Commons Recommended Citation Liu, Jiuqiang, "Maximal and Maximum Independent Sets in Graphs" (1992). Dissertations. 1985. https://scholarworks.wmich.edu/dissertations/1985 This Dissertation-Open Access is brought to you for free and open access by the Graduate College at ScholarWorks at WMU. It has been accepted for inclusion in Dissertations by an authorized administrator of ScholarWorks at WMU. For more information, please contact [email protected]. MAXIMAL AND MAXIMUM INDEPENDENT SETS IN GRAPHS by Jiuqiang Liu A Dissertation Submitted to the Faculty of The Graduate College in partial fulfillment of the requirements for the Degree of Doctor of Philosophy Department of Mathematics and Statistics Western Michigan University Kalamazoo, Michigan June 1992 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. MAXIMAL AND MAXIMUM INDEPENDENT SETS IN GRAPHS Jiuqiang Liu, Ph.D. Western Michigan University, 1992 A maximal independent set of a graph G is an independent set which is not contained properly in any other independent set of G. An independent set is called maximum if it is of largest cardinality. Denote i{G) to be the number of maximal independent sets of G. These special sets and the parameter i{G) have interested many researchers leading to a number of properties and results. One of these is the determination of the maximum number of maximal independent sets among all graphs of order n, and the extremal graphs. In this investigation,we develop new properties for the number of maximal independent sets i{G) and the number of maximum independent sets im(G), as well as determine the largest number of maximal and maximum independent sets possible in a ^-connected graph of order n(with n large) and characterize the respective extremal graphs. Finally, we determine the corresponding values for bipartite graphs and connected bipartite graphs, and characterize the extremal graphs. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. INFORMATION TO USERS This manuscript has been reproduced from the microfilm master. UMI films the text directly from the original or copy submitted. Thus, some thesis and dissertation copies are in typewriter face, while others may be from any type of computer printer. The quality of this reproduction is dependent upon the quality of the copy submitted. Broken or indistinct print, colored or poor quality illustrations and photographs, print bleedthrough, substandard margins, and improper alignment can adversely affect reproduction. In the unlikely event that the author did not send UMI a complete manuscript and there are missing pages, these will be noted. Also, if unauthorized copyright material had to be removed, a note will indicate the deletion. Oversize materials (e.g., maps, drawings, charts) are reproduced by sectioning the original, beginning at the upper left-hand corner and continuing from left to right in equal sections with small overlaps. Each original is also photographed in one exposure and is included in reduced form at the back of the book. Photographs included in the original manuscript have been reproduced xerographically in this copy. Higher quality 6" x 9" black and white photographic prints are available for any photographs or illustrations appearing in this copy for an additional charge. Contact UMI directly to order. UMI University Microfilms International A Bell & Howell Information Company 300 Norlti Zeeb Road. Ann Arbor. Ml 48106-1346 USA 313/761-4700 800/521-0600 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Order Number 9229686 Maximal and maximum Independent sets in graphs Liu, Jiuqiang, Ph.D. Western Michigan University, 1992 UMI 300 N. Zeeb Rd. Ann Arbor, MI 48106 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. To my loving wife Cong and my son Andrew Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. ACKNOWLEDGEMENTS I would like to thank Professor Yousef Alavi for his consistent encour­ agement and support throughout the entire period of my studies at Western Michigan University and for his supervision as my dissertation advisor. I am also grateful to Professor Paul Erdos, the outside referee of my dissertation, for his helpful suggestions and the many discussions on my disser­ tation topic. I also would like to thank Professor Allen Schwenk, my second reader, for reading the manuscript and offering valuable comments. I also wish to thank Professor Kenneth Williams, Professor Joseph Buckley and Professor Kung-Wei Yang for serving on my committee and for their help and support. Special thanks also go to my wife Cong for her love, support and under­ standing. Jiuqiang Liu 11 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. TABLE OF CONTENTS ACKNOWLEDGEMENTS ................................................................................. ii LIST OF FIGURES .............................................................................................. v CHAPTER I. INTRODUCTION ................................................................................. 1 1.1 Definitions and Notations .............................................. 1 1.2 History and Known Results .......................................... 2 II. CONSTRAINTS FOR INDEPENDENT SETS ............................. 7 2.1 Constraints for The Number of Maximal Independents Sets of a Graph .................... 7 2.2 Constraints for The Number of Maximum Independent Sets ......................................... 16 III. MAXIMAL INDEPENDENT SETS ................................................. 18 3.1 Maximal Independent Sets in ^-Connected Graphs ........................................................ 18 3.2 Maximal Independent Sets in Bipartite Graphs ............................................................. 39 3.3 Maximal Independent Sets in Connected Bipartite Graphs ......................................... 41 IV. MAXIMUM INDEPENDENT SETS AND OPEN PROBLEMS ................................................................... 56 m Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. Table of Contents - Continued CHAPTER 4.1 Maximum Independent Sets in fc-Connected Graphs ....................................................... 56 4.2 Open Problems ................................................................. 59 APPENDIX A.The Proof of Claim 2 .................................................................................... 60 REFERENCES....................................................................................................... 66 IV Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. LIST OF FIGURES 1.1 The Trees T„ ................................................................................................. 4 1.2 The Graph 6 2.1 A Graph G of Order n With i(G) = 0(3 3 )............................................ 13 3.1 The Graph ifj.n .......................................................................................... 18 3.2 The Graph Hz,n.......................................................................................... 19 3.3 The Graphs ff 4,n .......................................................................................... 20 3.4 The Graph ifs.Sr+i ..................................................................................... 21 3.5 Graphs G With A(G — u) = 1 ................................................................... 40 3.6 The Graphs Bgr, Where r > 3 and < > 2 ........................................... 44 3.7 Graphs of Order n W ith z{n) MIS’s...................................................... 45 3.8 The Graphs With Each Vertex of Degree > 3 in All Cycles ................. 46 3.9 The Graphs .BJr ........................................................................................... 48 3.10 The Graphs Q................................................................................................ 49 3.11 Graphs 0 W ith I 3 = T„ _ 4 ..................................................................... 51 3.12 Graphs O With G - u -N { u ) = ^ K 2 .............................................. 55 Reproduced with permission of the copyright owner. Further reproduction prohibited without permission. CHAPTER I INTRODUCTION 1.1 Definitions and Notations Let G be a graph. A subset V Ç V{G) is independent if no two vertices in V are adjacent. An independent set V is maximal if U' U {v} is not independent for any v E V{G) — V. A maximum independent set is an independent set of largest cardinality. The graph G is well-covered if every maximal independent set is also maximum (see [12]). A clique of G is a complete subgraph which is not contained properly in any other complete subgraph. Now, for a graph G, let I{G) be the set of maximal independent
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