D.Phil. (Mathematics) / Domination and its variants are now well studied in graph theory. One of these variants, paired-domination, requires that the subgraph induced by the dominating set contains a perfect matching. In this thesis, we further investigate the concept of paired-domination. Chapters 2, 3, 4, and 5 of this thesis have been published in [17], [41], [42], and [43], respectively, while Chapter 6 is under submission; see [44]. In Chapter 1, we introduce the domination parameters we use, as well as the necessary graph theory terminology and notation. We combine the de nition of a paired-dominating set and a locating set to de ne three new sets: locating-paired- dominating sets, di erentiating-paired-dominating sets, and metric-locating-paired- dominating sets. We use these sets in Chapters 3 and 4. In Chapter 2, we investigate the relationship between the upper paired-domination and upper total domination numbers of a graph. In Chapter 3, we study the properties of the three kinds of locating paired-dominating sets we de ned, and in Chapter 4 we give a constructive characterisation of those trees which do not have a di erentiating- paired-dominating set. In Chapter 5, we study the problem of characterising planar graphs with diameter two and paired-domination number four. Lastly, in Chap- ter 6, we establish an upper bound on the size of a graph of given order and paired- domination number and we characterise the extremal graphs that achieve equality in the established bound.
| Identifer | oai:union.ndltd.org:netd.ac.za/oai:union.ndltd.org:uj/uj:7689 |
| Date | 24 July 2013 |
| Creators | McCoy, John Patrick |
| Source Sets | South African National ETD Portal |
| Detected Language | English |
| Type | Thesis |
| Rights | University of Johannesburg |
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