Some Results On Graph Contraction Problems By Prafullkumar Prabhakar Tale MATH10201305002 The Institute of Mathematical Sciences, Chennai A thesis submitted to the Board of Studies in Mathematical Sciences In partial fulfillment of requirements for the Degree of DOCTOR OF PHILOSOPHY of HOMI BHABHA NATIONAL INSTITUTE May, 2020 STATEMENT BY AUTHOR This dissertation has been submitted in partial fulfillment of requirements for an advanced degree at Homi Bhabha National Institute (HBNI) and is deposited in the Library to be made available to borrowers under rules of the HBNI. Brief quotations from this dissertation are allowable without special permission, provided that accurate acknowledgement of source is made. Requests for permission for extended quotation from or reproduction of this manuscript in whole or in part may be granted by the Competent Authority of HBNI when in his or her judgement the proposed use of the material is in the interests of scholarship. In all other instances, however, permission must be obtained from the author. Prafullkumar Prabhakar Tale DECLARATION I hereby declare that the investigation presented in the thesis has been carried out by me. The work is original and has not been submitted earlier as a whole or in part for a degree / diploma at this or any other Institution / University. Prafullkumar Prabhakar Tale List of Publications arising from the thesis Journal 1. (Published) On the Parameterized Complexity of Contraction to Generalization of Trees : A. Agrawal, S. Saurabh, and P. Tale. Published in Theory of Computing Systems, Nov, 2018. Volume 63. p587-614. 2. (Under Review) Lossy Kernels for Graph Contraction Problems. : R. Krithika, P. Misra, A. Rai, and P. Tale. Under review in SIAM Journal on Discrete Mathematics (SIDMA). Conferences 1. Path Contraction Faster than 2n : A. Agrawal, F. Fomin, D. Lokshtanov, S. Saurabh and P. Tale. Accepted in 46th International Colloquium on Automata, Languages, and Programming, ICALP 2019. 2. An FPT Algorithm for Contraction to Cactus : R. Krithika, P. Misra, and P. Tale. Appered in Proc. of Annual International Computing and Combinatorics Conference, COCOON 2018. 3. Paths to Trees and Cacti. : A. Agrawal, L. Kanesh, S. Saurabh, and P. Tale. Appered in Proc. of Algorithms and Complexity - 10th International Conference, CIAC 2017. 4. On the Parameterized Complexity of Contraction to Generalization of Trees. : A. Agrawal, S. Saurabh, and P. Tale. Appered in Proc. of International Symposium on Parameterized and Exact Computation, IPEC 2017. 5. Lossy Kernels for Graph Contraction Problems. : R. Krithika, P. Misra, A. Rai, and P. Tale. Appered in Proc. of Foundations of Software Technology and Theoretical Computer Science, FSTTCS 2016. Prafullkumar Prabhakar Tale Dedicated to Vara Prasad, my brother-in-forest who dreamt of doing research in Physics while climbing in remote Himalayas. He never rested in peace, I doubt his soul will either. Acknowledgements In past few years, many people believed in me more than I believed myself. The foremost among them is my adviser, Saket Saurabh. I am grateful to him for all the time and energy he has spent over last five years educating me. I thank him for going out of his way to nourish me. His care goes beyond professional boundaries. He worked hard to create platforms where I can flurish. I will forever appreciate all the discussions, academics and otherwise, we had. He showed an extraordinary level of patience towards my academic flaws. His encouragements helped me to reach this stage. If I can cultivate one tenth of his work-ethics and ability for hard work, I will feel confident about my future. I am in debt of G Philip and V Raman for mentoring me on various projects. I am grateful to Philip for all the support he has provided in last couple of years. I would also like to thank Abhishek, Akanksha, Ashutosh, Daniel, Diptapriyo, Fahad, Fedor, Krithika, Lawqueen, Meesum, Pallavi, Pranabendu, Roohani, Sanjukta, Spoorthy, and Sudeshna for all the technical discussions we had. Those thoughtful discussions cemented my understanding of various topics and opened new directions for my work. I also like to thank other faculties at IMSc - V Arvind, Kamal Lodha, Meena Mahajan, R Ramanujam, Vikram Sharma, and C R Subramanian. My sincere thanks to Administrative, Library and Technical Staff for their help. I was fortunate to find supporting friends during my graduate studies. I will forever cherish moments shared with Aashita, Akanksha, Anantha, Anuj, Deeksha, Lawqueen, Krithika, Rajesh, Roohani, Sanjukta, Saumia, and Swaroop. My special thanks to Roohani for being there as a support system when I needed one. I would also like to thank all my other friends from school, college, and Chennai Trekking Club. Words fail me to describe my gratitude towards my parents and other family members. They had always been blind supporters of my decisions. Their faith in me is the immense source of inspiration to excel in the work I do. Contents Summary 16 List of Figures 19 1 Introduction 23 1.1 Preamble . 23 1.2 Known Results about Graph Contraction . 27 1.3 Scope of this thesis . 34 2 Preliminaries 39 2.1 Graph Theory . 39 2.2 Graph Contraction . 42 2.3 Parameterized Complexity . 44 2.4 Lossy Kernelization . 48 3 Tree Contraction 55 3.1 Introduction . 55 13 3.2 Preliminaries . 57 3.3 Kernel for BOUNDED TREE CONTRACTION ................ 61 3.4 Kernel Lower Bound for BOUNDED TREE CONTRACTION ........ 66 3.5 Lossy Kernel for TREE CONTRACTION .................. 70 3.6 Conclusion . 80 4 Cactus Contraction 83 4.1 Introduction . 83 4.2 Preliminaries . 85 4.3 Kernel for BOUNDED CACTUS CONTRACTION .............. 96 4.4 Kernel Lower Bound for BOUNDED CACTUS CONTRACTION ......106 4.5 Lossy Kernel for CACTUS CONTRACTION .................110 4.6 An FPT Algorithm for CACTUS CONTRACTION .............124 4.7 Conclusion . 154 5 Contraction to Generalization of Trees 157 5.1 Introduction . 157 5.2 Preliminaries . 158 5.3 Hardness results for T`-CONTRACTION ..................162 5.4 Lossy Kernel for T`-CONTRACTION ....................163 5.5 Randomized FPT Algorithm for T`-CONTRACTION ...........172 5.6 Derandomization of the FPT Algorithm . 185 5.7 Conclusion . 190 6 Out-Tree Contraction 191 6.1 Introduction . 191 6.2 Preliminaries . 192 6.3 Kernel for BOUNDED OUT-TREE CONTRACTION .............196 6.4 Kernel Lower Bound for BOUNDED OUT-TREE CONTRACTION .....200 6.5 Lossy Kernel for OUT-TREE CONTRACTION ...............205 6.6 Conclusion . 215 7 Clique Contraction 217 7.1 Introduction . 217 7.2 Preliminaries . 219 7.3 Lossy Kernel for CLIQUE CONTRACTION .................221 7.4 (No) Lossy Kernel for s-CLUB CONTRACTION ..............230 7.5 Conclusion . 234 8 Path Contraction 235 8.1 Introduction . 235 8.2 Preliminaries . 236 8.3 Enumeration of Connected Sets . 239 8.4 3-DISJOINT CONNECTED SUBGRAPH ...................242 8.5 Exact Algorithm for Path Contraction . 250 8.6 Conclusion . 268 Bibliography 279 Summary For a family of graphs F , the F -EDITING problem takes as an input a graph G and an integer k, and the objective is to decide if at most k edit operations on G can result in a graph that belongs to F . Various graph editing problems have been considered in the literature. These graph editing problems generalize many NP-Hard problems. Most of the studies regarding F -EDITING have been restricted to combination of vertex deletion, edge deletion or edge addition. Only recently, edge contraction as an edit operation has started to gain attention. The contraction of edge uv in graph G deletes vertices u and v from G, and replaces them by a new vertex, which is made adjacent to vertices that were adjacent to either u or v. In this thesis, we explore F -CONTRACTION for various graph classes from the viewpoints of parameterized complexity, lossy kernelization and exact algorithms.We extend the known boundaries about graph contraction problems in several ways. We consider F -CONTRACTION problems which do not have a polynomial kernels when parameterized by solution size. We compliment this negative result in two ways. Firstly, we present a polynomial kernel when parameterized by solution size and an additional parameter. In other words, we identify new graph classes for which there is a polynomial kernel. We also prove that these kernels are optimal under certain complexity conjecture. Secondly, we present a lossy kernel of polynomial size for all these problems. We present two FPT algorithms to append the list of graph classes F for which F -CONTRACTION parameterized by solution size is FPT. We end this thesis with a non-trivial exact algorithm 17 to determine what is the largest size of graph in a specific F to which an input graph can be contracted. To best of our knowledge, this is first such kind of algorithm in case of graph contraction problems. List of Figures 2.1 Graph contraction operation . 43 3.1 Operation SPLIT(T,v,L,R) with L = x and R = x ,x .. 58 { 3} { 1 2} 3.2 Modifying big witness sets which are leafs. All but one vertex in W(ti) has been moved to W(t j). See Observation 3.2.3. 59 3.3 An illustration of Reduction Rule 3.3.1. 61 3.4 Parts of a longest path from root to a leaf. See Lemma 3.3.2. 64 3.5 Kernel lower bound for BOUNDED TC. .................. 66 3.6 Partition of input graph. Please see Reduction 3.5.3 . 75 3.7 Please refer to Lemma 3.5.5 . 77 4.1 Cactus graph and its block decomposition. 86 4.2 Operation SPLIT(T,v,L,R) with L = w,x and R = x ,x ....... 90 { 3} { 1 2} 4.3 An illustration of Reduction Rule 4.3.3. 100 4.4 Kernel lower bound for BOUNDED CC.