Ph.D. / In this thesis, our primary objective is to investigate the effects that various graph modifications have on the total domination number of a graph. In Chapter 1, we introduce basic graph theory concepts and preliminary definitions. In Chapters 2 and 3, we investigate the graph modification of edge removal. In Chapter 2, we characterize graphs for which the removal of any arbitrary edge increases the total domination number. We also begin the investigation of graphs for which the removal of any arbitrary edge has no effect on the total domination number. In Chapter 3, we continue this investigation and determine the minimum number of edges required for these graphs. The contents of Chapters 2 and 3 have been published in Discrete Applied Mathematics [15] and [16]. In Chapter 4, we investigate the graph modification of edge addition. In particular, we focus our attention on graphs for which adding an edge between any pair of nonadjacent vertices has no effect on the total domination number. We characterize these graphs, determine a sharp upper bound on their total domination number and determine which combinations of order and total domination number are attainable. 10 11 We also study claw-free graphs which have this property. The contents of this chapter were published in Discrete Mathematics [20]. In Chapter 5, we investigate the graph modification of vertex removal. We characterize graphs for which the removal of any vertex changes the total domination number and find sharp upper and lower bounds on the total domination number of these graphs. We also characterize graphs for which the removal of an arbitrary vertex has no effect on the total domination number and we further show that they have no forbidden subgraphs. The contents of this chapter were published in Discrete Applied Mathematics [14]. In Chapters 6 and 7, we investigate the graph modification of edge lifting. In Chapter 6, we show that there are no trees for which every possible edge lift decreases the domination number, and we characterize trees for which every possible edge lift increases the domination number. The contents of Chapter 6 were published in the journal Quaestiones Mathematicae [17]. In Chapter 7, we show that there are no trees for which every possible edge lift decreases the total domination number and that there are no trees for which every possible edge lift leaves the total domination number unchanged. We characterize trees for which every possible edge lift increases the total domination number. At the time of the writing of this thesis, the contents of Chapter 7 have been published online in the Journal of Combinatorial Optimization [18] and will appear in print in a future issue.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uj/uj:2717 |
| Date | 20 August 2012 |
| Creators | Desormeaux, Wyatt Jules |
| Source Sets | South African National ETD Portal |
| Detected Language | English |
| Type | Thesis |
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