ZERO FORCING AND POWER DOMINATION IN GRAPHS SUDEEP STEPHEN Master of Science (Mathematics)(University of Madras) School of Mathematical and Physical Sciences The University of Newcastle, Australia A thesis submitted in total fulfilment of the requirements for the degree of Doctor of Philosophy October 2017 Statements Statement of Originality The thesis contains no material which has been accepted for the award of any other degree or diploma in any university or other tertiary institution and, to the best of my knowledge and belief, contains no material previously published or written by another person, except where due reference has been made in the text. I give consent to the final version of my thesis being made available worldwide when deposited in the University's Digital Repository, subject to the provisions of the Copyright Act 1968. Statement of Authorship I hereby certify that the work embodied in this thesis contains published papers of which I am a joint author. I have included as part of the thesis a written statement, endorsed by my supervisor, attesting to my contribution to the joint publication. Signed: SUDEEP STEPHEN The candidate has six papers, of which he is a joint author, embodied in the thesis. As principal supervisor I can attest that the candidate's contribution, in all cases were at least 80% of the final article. Signed: Dr JOE RYAN iv List of Publications Publications Arising from this Thesis (1) D. Ferrero, T. Kalinowski and S. Stephen. Zero forcing in iterated line digraphs. Submitted to Discrete Applied Mathematics, 2017 [37]. (2) R. Davila, T. Kalinowski and S. Stephen. Proof of a conjecture of Davila and Kenter regarding a lower bound for the forcing number in terms of girth and minimum degree. Submitted to Discrete Applied Mathematics, 2016 [23]. (3) D. Ferrero, C. Grigorious, T. Kalinowski, J. Ryan and S. Stephen. Minimum rank and zero forcing number for butterfly networks. Submitted to Discrete Applied Mathematics, 2016 [38]. (4) C. Grigorious, T. Kalinowski, J. Ryan and S. Stephen. On the power domination number of de Bruijn and Kautz digraphs. To appear in Lecture Notes in Computer Science, 2017 [45]. (5) S. Stephen, B. Rajan, C. Grigorious and A. William. Resolving-power dominating sets. Applied Mathematics and Computation, 256, 778{785, 2015 [85]. (6) S. Stephen, B. Rajan, J. Ryan, C. Grigorious and A. William. Power domination in certain chemical structures. Journal of Discrete Algorithms, 33, 10{18, 2015 [86]. Other Publications Produced During my Candidature (1) C. Grigorious, T. Kalinowski, J. Ryan and S. Stephen. Metric dimension of Circulant graph C(n; ±{1; 2; 3; 4g). The Australasian Journal of Combinatorics, 69 (3), 417{ 441, 2017 [46]. (2) C. Grigorious, S. Stephen, B. Rajan and M. Miller. On the Partition Dimension of Circulant Graphs. The Computer Journal, 60 (2), 180{184, 2016 [48]. (3) S. Klavˇzar, P. Manuel, M. J. Nadjafi-Arani, R. S. Rajan, C. Grigorious and S. Stephen. Average Distance in Interconnection Networks via Reduction Theorems for Vertex-Weighted Graphs. The Computer Journal, 59(12), 1900{1910, 2016 [68]. (4) C. Grigorious, P. Manuel, M. Miller, B. Rajan and S. Stephen. On the metric dimension of circulant and Harary graphs. Applied Mathematics and Computation, 248, 47{54, 2014 [47]. (5) P. Manuel, B. Rajan, C. Grigorious and S. Stephen. On the Strong Metric Dimen- sion of Tetrahedral Diamond Lattice. Mathematics in Computer Science, 9, 201{208, 2015 [76]. (6) C. Grigorious, S. Stephen, B. Rajan, M. Miller and A. William. On the Partition Dimension of a Class of Circulant Graphs. Information Processing Letters, 114, 353{ 356, 2014 [49]. To my parents, Stephen P.M and Mary Stephen And in memory of my most beloved supervisor, Mirka Miller. Acknowledgements First and foremost, I would like to express my sincere gratitude to my su- pervisors, (L) Prof. Mirka Miller, Dr. Joe Ryan, Dr. Thomas Kalinowski and Dr. Paul Manuel, who have supported me with their encouragement, guidance and patience. My sincere gratitude to Mirka and Joe, without them, I would never have been in Newcastle. Thank you for trusting my research capabilities and above all for the emotional and practical support whenever I needed them. Thank you for teaching how to build collaborations and introducing me to all your collaborators. I can never forget the good times we had spent together. Your motivations and encouragements helped me to be a better person and a researcher. It was indeed a tough phase for me and Joe after Mirka lost her battle with cancer. Thank you Joe for being there always and guiding me through the completion of my candidature. I am happy we did this together for Mirka. I specially like to acknowledge Thomas Kalinowski for his insightful sug- gestions and comments on my research and talks at every stage of my Ph.D. I am always inspired by his work. To be very honest, I have always looked up at him as my role model. I also thank Dr. Paul Manuel for contributing 25% towards my scholarship. My special thanks to Dr. Bharati Rajan, my mentor, for introducing me to graph theory and research. I owe what I am today to you. I am thankful to Dr. Cristina Dalfo Simo, my external supervisor for her inputs and support during my Ph.D candidature. I am grateful to Prof. Indra Rajasingh and Dr. Sundara Rajan for their encouragement and support during my initial stages of research in Chennai. Discussions and chats with Prof. Brian Alspach were always interesting and helpful. Thank you. I am also thankful to Renee Mam and the entire Maths department of St. Joseph's College, Bangalore for inspiring me to do research in mathematics. I like to thank Prof. Zdenek Ryacek and Prof. Dafik for hosting me during my stay in The University of West Bohemia, Czech Republic and University of Jember, Indonesia respectively. I also like to thank my col- laborators Randy Davila, Prof. Daniela Ferrero and Prof. Sandi Klavzar for accepting my request to work with me and sharing their knowledge. It was indeed wonderful 3 years in Newcastle. I would like to acknowledge Dr. Benny Sudakov for pointing out the argument for Theorem 5.2.3 on his visit to the University of Newcastle during the 40 ACCMCC. Thanks to the University of Newcastle for providing scholarships (UNIPRS and UNRSC 50-25) to pursue my dream. I thank the International Leader- ship Experience and Development (ILEAD) Plus Program for the generous support for participating in various conferences. I acknowledge the Faculty of Sciences for the Conference scholarship. My sincere thanks to the school of mathematical and physical sciences. All the staff, both academic and administrative staff created a healthy and comfortable environment for me to pursue my research. Many thanks to all of them for their supports and efforts. Special thanks to Vicki, Anna Rosa, Irene, Rosemary and Felicity for their administrative help whenever I needed them. To my friend and colleague Cyriac Grigorious, thank you so much for being there with me for the past 8 years. You have influenced my life more than you can imagine. Thank you for the support and friendship during my stay here. To my office mates, Bong, Dushyant, Oudone and Wulan, thank you for bearing me. Thanks to the Malayali community in Newcastle and Sydney for making me welcome among them and making me feel at home away from home. I am indebted to my parents, Stephen P. M and Mary Stephen, my wife, Jinu Joseph, my brother, Suraj Stephen, my sister-in-law, Rakhi Ashok and my nephew, Aswin Suraj, for their love and support during these years. A special word of acknowledgement to Mr. Simon, Saly amme, Ashwin and Ada for considering me a part of their family. Thank you for the unconditional love you have shown towards me. Saly amme, thank you so much for being there like a mother encouraging, supporting and guiding me through various frustrations and disappointments. Last but not least, I thank God Almighty for blessing me with this opportunity to pursue my research and for giving me sound body and mind to complete this successfully. Sudeep Stephen Contents List of Figures xi Nomenclature xv Chapter 1. Introduction1 1.1. Contributions of the thesis3 1.2. Structure of the thesis3 Chapter 2. Background Theory5 2.1. Graph theory5 2.1.1. Undirected graphs5 2.1.2. Graph operations8 2.1.3. Directed graphs 10 2.2. Zero forcing in graphs 11 2.2.1. Bounds on the zero forcing number 17 2.2.2. Zero forcing in directed graphs 21 2.2.3. Variants of zero forcing problem 23 2.3. Power domination in graphs 24 2.3.1. Bounds on the power domination number 26 2.3.2. Power domination in directed graphs 31 2.3.3. Variants of power domination problem 32 2.4. Zero forcing and power domination 33 Chapter 3. Zero forcing and power domination in iterated line digraphs 35 3.1. An alternate interpretation 35 3.2. Zero forcing and power domination in de Bruijn digraphs 38 3.3. Zero forcing and power domination number in Kautz digraphs 41 3.4. Zero forcing in iterated line digraphs 45 3.5. Power domination in iterated line digraphs 48 3.6. Applications 52 Chapter 4. Zero forcing and power domination in butterfly networks 55 4.1. An alternate interpretation 55 4.2. Zero forcing in butterfly network 57 ix x CONTENTS 4.2.1. The upper bound for Z(BF(r)) 58 4.2.2. The lower bound for Z(BF(r)) 62 4.3. Power domination in butterfly network 71 Chapter 5. Zero forcing number in terms of girth and minimum degree 77 5.1.