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Irreversible K-Threshold Conversion Processes on Graphs by Jane Irreversible k-Threshold Conversion Processes on Graphs by Jane Wodlinger B.Sc., University of Victoria, 2009 M.Sc., Simon Fraser University, 2012 A Dissertation Submitted in Partial Fulfillment of the Requirements for the Degree of DOCTOR OF PHILOSOPHY in the Department of Mathematics and Statistics © Jane Wodlinger, 2018 University of Victoria All rights reserved. This dissertation may not be reproduced in whole or in part, by photocopying or other means, without the permission of the author. ii Irreversible k-Threshold Conversion Processes on Graphs by Jane Wodlinger B.Sc., University of Victoria, 2009 M.Sc., Simon Fraser University, 2012 Supervisory Committee Dr. C. M. Mynhardt, Supervisor (Department of Mathematics and Statistics) Dr. P. J. Dukes, Departmental Member (Department of Mathematics and Statistics) Dr. V. Srinivasan, Outside Member (Department of Computer Science) iii Supervisory Committee Dr. C. M. Mynhardt, Supervisor (Department of Mathematics and Statistics) Dr. P. J. Dukes, Departmental Member (Department of Mathematics and Statistics) Dr. V. Srinivasan, Outside Member (Department of Computer Science) ABSTRACT Given a graph G and an initial colouring of its vertices with two colours, say black and white, an irreversible k-threshold conversion process on G is an iterative process in which a white vertex becomes permanently coloured black at time t if at least k of its neighbours are coloured black at time t − 1. A set S of vertices is an irreversible k-threshold conversion set (k-conversion set) of G if the initial colouring in which the vertices of S are black and the others are white results in the whole vertex set becoming black eventually. In the case where G is (k + 1)-regular, it can be shown that the k-conversion sets coincide with the so-called feedback vertex sets, or decycling sets. In this dissertation we study the size, ck(G), and structure of minimum k-conversion sets in several classes of graphs, G. We examine conditions that lead to equality and inequality in existing bounds on ck(G) for k- and (k+1)-regular graphs. Furthermore, we derive new sharp lower bounds on ck(G) for regular graphs of degree ranging from k + 1 to 2k − 1 and for graphs of maximum degree k + 1. We determine exact values of ck(G) for certain classes of trees. We show that every (k + 1)-regular graph has a minimum k-conversion set that iv avoids certain structures in its induced subgraph. These results lead to new proofs of several known results on colourings and forest partitions of (k + 1)-regular graphs and graphs of maximum degree k + 1. v Contents Supervisory Committee ii Abstract iii Table of Contents v List of Tables vii List of Figures viii 1 Introduction 1 1.1 Outline . 4 2 Background and preliminaries 6 2.1 k-immune sets . 9 2.2 The k-conversion number . 10 2.2.1 Exact values (and near-exact values) . 11 2.2.2 Bounds . 18 2.3 Decycling sets, induced forests and k-conversion in (k +1)-regular graphs 24 2.3.1 Lower bounds on the decycling number . 26 2.3.2 Upper bounds on the decycling number . 27 2.3.3 The decycling number of hypercubes . 31 2.3.4 Cubic graphs that meet the lower bound . 33 2.3.5 Interpolation . 35 2.4 Complexity and algorithmic results . 36 3 The k-conversion number of regular graphs 38 3.1 The k-conversion number of k-regular graphs . 38 3.2 The k-conversion number of (k + 1)-regular graphs . 40 vi 3.2.1 k-conversion sets of size k in (k + 1)-regular graphs . 40 3.2.2 A lower bound on ck(G) for (k + 1)-regular graphs . 43 3.3 A lower bound on ck(G) for (k + r)-regular graphs . 45 4 The 2-conversion number of cubic graphs 48 4.1 Generalized fullerenes . 48 4.2 Snarks and would-be snarks . 55 4.3 3-edge connected cubic graphs . 64 5 A lower bound on ck(G) for graphs of maximum degree k + 1 73 5.1 Definitions and preparation . 73 5.2 The lower bound . 83 5.2.1 Graphs for which the lower bound is sharp . 85 5.2.2 Graphs of maximum degree k + 1 with large k-conversion number 86 6 k-conversion in trees 88 6.1 Caterpillars . 89 6.2 Spiders and double spiders . 91 7 Subgraph-avoiding minimum k-conversion sets in (k + 1)-regular graphs 94 7.1 Independent minimum 2-conversion sets in cubic graphs . 96 7.2 Possible generalizations of Theorem 7.2 . 102 7.3 Avoiding Kk in minimum k-conversion sets of (k + 1)-regular graphs . 104 7.4 Acyclic minimum 3-conversion sets in 4-regular graphs . 109 7.5 Avoiding (k−1)-regular subgraphs in k-conversion sets of (k+1)-regular graphs . 115 7.6 Consequences of Theorems 7.2, 7.13, 7.19 and 7.24 . 122 8 Conclusion and open problems 131 Bibliography 135 vii List of Tables 4.1 Combinations of snark properties that permit equality/inequality in the lower bound on c2(G).......................... 60 viii List of Figures 2.1 A 2-conversion process. .7 2.2 A minimum 4-conversion set (shown in black) in a 7 × 5 toroidal grid. 15 2.3 The Cartesian product of two graphs. 18 2.4 A 2-conversion set of the Cartesian product shown in Figure 2.3 as constructed in the proof of Theorem 2.18. 20 2.5 The tensor product of two graphs. 20 2.6 A 2-conversion set of the tensor product shown in Figure 2.5 as con- structed in the proof of Theorem 2.20. 22 2.7 A graph achieving equality in the bounds of Proposition 2.27. 28 2.8 The graphs G1 and G2 of Theorem 2.29. 29 3.1 A 5-regular graphs with a 5-conversion set of size 5. 39 3.2 A graph that achieves equality in the bound of Proposition 3.6. 43 4.1 Fullerenes. 49 4.2 A 4 by 8 cylindrical grid. 50 4.3 An alternative picture of a generalized fullerene. 52 4.4 A minimum 2-conversion set in the Petersen graph. 56 4.5 The flower snark J5.............................. 57 4.6 Building blocks for graphs that exceed the bound. 61 4.7 The triangle-replaced graph of the Petersen graph. 63 − 4.8 An example of the construction of a cubic graph G ○ A ........ 66 5.1 A graph from which it is impossible to obtain a simple ∆-regular graph by adding a forest. 75 5.2 Illustration of the construction of a ∆-regular graph from a graph of maximum degree ∆. 82 5.3 The construction of the ∆-regular graph G′ from the proof of Theo- rem 5.10. 84 ix 5.4 A subcubic graph for which c2(G) exceeds the lower bound. 86 6.1 Type 2 minimal 2-immune sets and (k; k + 1)-paths in a caterpillar. 90 6.2 A spider and a double spider. 91 7.1 Illustration of seed shuffling for k = 2.................... 95 7.2 The stopping condition for the algorithm from the proof of Theorem 7.2. ....................................... 101 7.3 A graph in G3................................. 103 7.4 The sequence of Kk's in the proof of Theorem 7.13. 109 7.5 The sequence of cycles in the proof of Theorem 7.19 (Case 2). 114 7.6 The outcome of the algorithm in the proof of Theorem 7.24. 123 Chapter 1 Introduction Network diffusion processes have been studied by social scientists since the mid- twentieth century to understand how ideas and information proliferate through com- munities, and how new technologies and behaviours become widely adopted. These studies set out to quantify, and often model, the extent to which people's behaviour and decisions are influenced by their peers [59, 87]. For example, an early network dif- fusion study aiming to explain the rapid acceptance of hybrid corn seed by Iowa farm- ers in the 1930s found that the most influential factor leading to farmers' acceptance of the new seed was their neighbours' attitudes towards it [96]. Other early studies an- alyzed the adoption of new drugs and medical technologies by physicians, examining the peer-to-peer transmission of attitudes towards the new treatments [31, 112]. The theory of network diffusion has obvious applications in marketing [24, 49, 59, 73], especially with the rise of viral marketing strategies, and in the context of public health campaigns. A company wishing to market its product efficiently may do so by identifying influential people in the network and, for example, giving them the product for free. A natural problem, then, is to find a small set of people whose adoption of the product will lead to the desired level of market saturation [26] (this is often called the target set selection problem [5]). Other applications of network diffusion include epidemiology and disease con- trol [4, 38], game theory [78, 115], and the design of computer networks and power grids [13, 38, 117]. Everett Rogers was the first to synthesize a large number of dif- 2 fusion studies from many disciplines into a common theory on the adoption of new innovations. His book Diffusion of Innovations [95], originally published in 1962 and now in its fifth edition, provides a comprehensive reference on the topic. Unsurprisingly, network diffusion processes are highly dependent on the structure of the network in question, which is naturally modelled by a graph. To model the diffusion process, each vertex is in one of two states (say black or white) at time t and its state at time t + 1 is determined, according to some conversion rule, by the states of its neighbours at time t.
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