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Research Topics in Graph Theory and Its Applications Research Topics in Graph Theory and Its Applications Research Topics in Graph Theory and Its Applications By Vadim Zverovich Research Topics in Graph Theory and Its Applications By Vadim Zverovich This book first published 2019 Cambridge Scholars Publishing Lady Stephenson Library, Newcastle upon Tyne, NE6 2PA, UK British Library Cataloguing in Publication Data A catalogue record for this book is available from the British Library Copyright © 2019 by Vadim Zverovich All rights for this book reserved. No part of this book may be reproduced, stored in a retrieval system, or transmitted, in any form or by any means, electronic, mechanical, photocopying, recording or otherwise, without the prior permission of the copyright owner. ISBN (10): 1-5275-3533-9 ISBN (13): 978-1-5275-3533-6 Disclaimer: Any statements in this book might be fictitious and they represent the author's opinion. In memory of my son, Vladik (1987 { 2000) . Contents Preface xi 1 α-Discrepancy and Strong Perfectness 1 1.1 Background and Aims . 1 1.2 α-Discrepancy . 4 1.3 Strongly Perfect Graphs . 7 1.4 Computational Aspects . 10 1.5 Beneficiaries and Dissemination . 13 1.6 References . 14 1.7 Resources and Work Plan . 16 1.8 Reviewers' Reports . 19 1.9 PI's Response to Reviewers' Comments . 30 1.10 Summary of Key Lessons . 32 2 Braess' Paradox in Transport Networks 35 2.1 Background and Aims . 35 2.2 Braess' Paradox and Methodology . 40 2.3 Computational Aspects of Braess' Paradox . 44 2.4 Braess-like Situations in Rail Networks . 48 2.5 References . 49 2.6 Beneficiaries, Resources and Work Plan . 50 2.7 Impact Statement . 53 2.8 Letters of Support . 57 2.9 Reviewers' Reports . 59 2.10 PI's Response to Reviewers' Comments . 71 vii viii CONTENTS 2.11 Summary of Key Lessons . 76 3 Hazard Assessment of Infrastructures for Smart Evacuation Based on Unified Networks 79 3.1 Background . 79 3.2 National Importance . 84 3.3 Academic Impact . 85 3.4 Research Idea and Objectives . 86 3.5 Programme and Methodology . 88 3.6 References . 94 3.7 Resources and Work Plan . 96 3.8 Impact Statement . 99 3.9 Reviewers' Reports . 102 3.10 PI's Response to Reviewers' Comments . 115 3.11 Summary of Key Lessons . 119 4 The Probabilistic Method for Domination Parameters 123 4.1 Aims and Background . 123 4.2 The Classical Bound and Related Results . 124 4.3 Proof Technique of the Conjectures . 127 4.4 Generalisation of the Conjecture for Parametric Domination . 130 4.5 References . 131 4.6 Resources and Work Plan . 132 4.7 Reviewers' Reports . 133 4.8 PI's Response to Reviewers' Comments . 138 4.9 Summary of Key Lessons . 140 5 Decomposition Methods for the Reconstruction Conjectures 143 5.1 Background and Aims . 143 5.2 Reconstruction and Decompositions . 146 5.3 Reconstructible Graphs Admitting Decompositions . 150 5.4 References . 153 CONTENTS ix 5.5 Resources and Work Plan . 155 5.6 Reviewers' Reports . 157 5.7 PI's Response to Reviewers' Comments . 165 5.8 Summary of Key Lessons . 166 6 Embedding Graphs on Topological Surfaces 169 6.1 Aims and Background . 169 6.2 Devising an Implementable Polynomial-Time Toroidality Testing Algorithm . 173 6.3 Structural Properties of Graphs Embeddable on Surfaces . 177 6.4 Enumeration and Automorphism Groups . 178 6.5 Beneficiaries, Applications and Dissemination . 179 6.6 Programme of Work . 180 6.7 References . 183 6.8 Resources . 186 6.9 Reviewers' Reports . 188 6.10 PI's Response to Reviewers' Comments . 195 6.11 Summary of Key Lessons . 197 7 Development and Mining of a Comprehensive Database of Graph Invariants 199 7.1 Aims, Objectives and Methodology . 199 7.2 Development of the Database . 203 7.3 Mining of the Database . 206 7.4 References . 211 7.5 Resources and Work Plan . 217 7.6 Reviewers' Reports . 219 7.7 PI's Response to Reviewers' Comments . 225 7.8 Summary of Key Lessons . 225 8 Graph-Theoretic Conjectures and the Computer System GRAPHOGRAPH 227 8.1 Aims and Background . 227 8.2 Graph Polynomials and the Unimodal Conjectures . 229 x CONTENTS 8.3 Domination Parameters of Graphs . 231 8.4 Conjectures Generated by Graffiti . 232 8.5 GRAPHOGRAPH and Graph-Theoretic Properties . 234 8.6 References . 236 8.7 Beneficiaries, Resources and Work Plan . 239 8.8 Reviewers' Reports . 242 8.9 PI's Response to Reviewer 1's Comments . 248 8.10 Summary of Key Lessons . 250 9 Man-Machine Approach to Investigation into Hereditary Classes of Graphs 253 9.1 Aims and Background . 253 9.2 Applying Software for Preliminary Analysis of the Classes . 255 9.3 Formulating Conjectures . 256 9.4 Proving Conjectures . 258 9.5 References . 263 9.6 Resources and Work Plan . 266 9.7 Reviewers' Reports . 268 9.8 PI's Response to Reviewers' Comments . 276 9.9 Summary of Key Lessons . 278 List of Abbreviations 281 Index 283 Preface This book includes a number of research topics in graph the- ory and its applications. The topics are in the form of research projects developed by the author over the last 15 years. We dis- cuss various research ideas devoted to α-discrepancy, strongly perfect graphs, the reconstruction conjectures, graph invariants, hereditary classes of graphs, embedding graphs on topological surfaces, as well as applications of graph theory, such as trans- port networks and hazard assessments based on unified net- works. In addition to the original research ideas presented and methods to address them, there are also examples of impact statements, project resources required, support letters and work plans. The book has a free-form structure that allows the reader freedom, that is, the chapters are independent and can be read in any order. Another important feature is the inclusion of reviewers' opin- ions in each chapter, which outline the strengths and weaknesses of various aspects of multiple research projects. The viewpoints of reviewers will be useful for recognising the typical mistakes authors make in research proposals. This book is ideal for de- velopers of grant proposals, as well as for researchers interested in exploring new areas of graph theory and its applications. Ad- vanced students in graph theory may use the topics presented in this book to develop their final-year projects, master's theses or doctoral dissertations. It is the author's hope that this publication of original re- search ideas, problems and conjectures will instigate further re- xi xii PREFACE search, or even a resurgence of interest, in the aforementioned important areas of graph theory. I am very grateful to my wife and two daughters for their patience and support during the completion of this book. Chapter 1 α-Discrepancy and Strong Perfectness 1.1 Background and Aims Discrepancy theory, which originated from number theory, can generally be described as the study of irregularities of distribu- tions in various settings. Classical combinatorial discrepancy theory is devoted to addressing the problem of partitioning the vertex set of a hypergraph into two classes in such a way that all hyperedges are split into approximately equal parts by the classes. That is, it is devoted to measuring the deviation of an optimal partition from a perfect partition when all hyper- edges are split into equal parts. It should be noted that many classical results in various areas of mathematics (e.g. geometry and number theory) can be formulated in such terms. Combi- natorial discrepancy theory was introduced by Beck in [3] and studied in [2]{[5] and [23]. F¨urediand Mubayi [12] indicated that discrepancy theory had \developed into an elaborate field related ::: to geometry, probability theory, ergodic theory, com- puter science, and combinatorics", while Tezuka [26] described its application to finance. Among practical applications of the theory are image pro- 1 2 CHAPTER 1. α-DISCREPANCY & STRONG PERFECTNESS cessing and the Monte Carlo methods for high dimensions. An important role in discrepancy theory is played by the fundamen- tal \six-standard-deviation" result [25] and the discrepancy con- jecture [4]. An interesting version of discrepancy is α-discrepan- cy, which occurs when success is measured by minimizing the imbalance of the vertex set while keeping the imbalance of each hyperedge at least α. The basic results in this context are de- voted to 1-discrepancy and are concentrated on upper bounds and the F¨uredi{Mubayi conjecture [12]. The development of graph theory over the last five decades has been strongly influenced by the Strong Perfect Graph Con- jecture and perfect graphs introduced by Berge in the early 1960s [6]. Perfect graphs are a fundamental concept in graph theory. This class of graphs has interesting applications, and there are books entirely devoted to perfect graphs (e.g. [7, 13]). The famous Strong Perfect Graph Conjecture, stated by Berge, had been open for about 40 years. Various attempts to prove it gave rise to many powerful methods, important concepts and interesting results in graph theory. Some of those methods af- fected the development of the theory of modular decomposition and Fulkerson's theory of antiblocking polyhedra. Chudnovsky, Robertson, Seymour and Thomas [9] relatively recently proved the Strong Perfect Graph Conjecture on 179 pages. In 1978 at a Monday Seminar in Paris, Berge introduced another important class of graphs called strongly perfect graphs. It is a subclass of perfect graphs (see [8]). A graph is strongly perfect if every induced subgraph contains an independent set that meets all maximal cliques. The known results on strongly perfect graphs can be found in the survey paper [22]. Unlike perfect graphs, strongly perfect graphs do not have a conjecture similar to the Strong Perfect Graph Conjecture.
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