Thesis (Ph.D.) - University of St Andrews, June 2010.

Thesis (M. Phil.)--University of Hong Kong, 1999.

Thesis (M.S.)--University of Florida, 2005. / Title from title page of source document. Document formatted into pages; contains 61 pages. Includes vita. Includes bibliographical references.

Thesis (M.Phil.)--Hong Kong University of Science and Technology, 2006. / Includes bibliographical references (leaves 56-58). Also available in electronic version.

Thesis (Ph.D.)--University of Nebraska-Lincoln, 2007. / Title from title screen (site viewed Feb. 17, 2009). PDF text: viii, 383 p. : ill. (chiefly col.) ; 49.9 Mb. UMI publication number: AAT 3294904. Includes bibliographical references. Also available in microfilm and microfiche formats.

Ph.D. / In Chapter 1 we will give a brief historical account of domination theory and define the necessary concepts which we use in the remainder of the thesis. In Chapter 2 we establish a lower bound for the minus k-subdomination number of trees and characterize those trees which achieve this lower bound. We also compute the value of Yks-101 for comets and for cycles. We then show that the decision problem corresponding to the computation of Yks-101 is NP-complete, even for bipartite graphs. In Chapter 3 we characterize those trees T which achieve the lower bound of Cockayne and Mynhardt, thus generalizing the results of [11] and [2]. We also compute Yks-11 for comets and cycles. In Chapter 4 we study the partial signed domination number of a graph. In particular, we establish a lower bound on Yc/d for regular graphs and prove that the decision problem corresponding to the computation of the partial signed domination number is NP-complete. Chapter 5 features the minus bondage number b- (G) of a nonempty graph G, which is defined as the minimum cardinality of a set of edges whose removal increases the minus domination number of G. We show that the minus bondage and ordinary bondage numbers of a graph are incomparable. Exact values for certain well known classes of graphs are computed and an upper bound for b- is given for trees. Finally, we show that the decision problem corresponding to the computation of b- is N P - hard, even for bipartite graphs. We conclude, in Chapter 6, by discussing possible directions for future research.

M.Sc. / Let G be a graph and D a set of vertices such that every vertex in G is in D or adjacent to at least one vertex in D. Then D is called a dominating set of G and the smallest cardinality of such a dominating set of G is known as the domination number of G, denoted by y(G). This short dissertation is a study of the domination number in graphs with bounds on both the minimum and maximum degrees. In Chapter 1 we give all definitions, terminology and references related to the material presented in this thesis. In Chapter 2 we study an article by McCuaig and Shepherd which considers graphs with minimum degree two and gives an upper bound for their domination numbers in terms of their order. This bound is also an improvement of one originally determined by Ore. In Chapter 3 an article by Fisher, Fraughnaugh and Seager is studied. Here the domination number in graphs with maximum degree at most three is discussed. Furthermore au upper bound on the domination number of a graph is given in terms of its order, size and the number of isolated vertices it contains. This result is an extension of a previous result by Reed on domination in graphs with minimum degree three. A set U of vertices of a graph G = (V, E) is k-dominating if each vertex of V — U is adjacent to at least k vertices of U. The k-domination number of G, Yk (G), is the smallest cardinality of a k-dominating set of G. Finally in Chapter 4 we study an article by Cockayne, Gamble and Shepherd which gives an upper bound for the k-domination number of a graph with minimum degree at least k. This result is a generalization of a result by Ore.

"This paper is concerned with systems of distinct representatives (abbreviated by S.D.R.) and related combinatorial topics"--Introduction. / Typescript. / "August, 1959." / "Submitted to the Graduate Council of Florida State University in partial fulfillment of the requirements for the degree of Master of Science." / Advisor: Marion F. Tinsley, Professor Directing Paper. / Includes bibliographical references (leaf 25).

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