D.Phil. / Domination in graphs is now well studied in graph theory and the literature on this subject has been surveyed and detailed in the two books by Haynes, Hedetniemi, and Slater [45, 46]. In this thesis, we continue the study of domination, by adding to the theory; improving a number of known bounds and solving two previously published conjectures. With the exception of the introduction, each chapter in this thesis corresponds to a single paper already published or submitted as a journal article. Despite the seeming disparity in the content of some of these articles, there are two overarching goals achieved in this thesis. The rst is an attempt to partition the vertex set of a graph into two sets, each holding a speci c domination-type property. The second is simply to improve known bounds for various domination parameters. In particular, an edge weighting function is presented which has been useful in providing some of these bounds. Although the research began as two separate areas of focus, there has been a fair degree of overlap and a number of the results contained in this thesis bridge the gap quite pleasingly. Specially, Chapter 11 uses the edge weighting function to prove a bound on one of the sets in our most fundamental partitions, while the improvement on a known bound presented in Chapter 7 was inspired by considering the possible existence of another partition. This latter proof relies implicitly on the `almost' existence of such a partition.

A graph G is 2-stratified if its vertex set is partitioned into two classes (each of which is a stratum or a color class.) We color the vertices in one color class red and the other color class blue. Let F be a 2-stratified graph rooted at some blue vertex v. The F-domination number γ F(G) of a graph G is the minimum number of red vertices of G in a red-blue coloring of the vertices of G such that every blue vertex v of G belongs to a copy of F rooted at v. In this paper we investigate the F-domination number for all 2-stratified graphs F of order n≤3 rooted at a blue vertex.

A set S of vertices in a graph G is a total dominating set if every vertex of G is adjacent to some vertex in S. The minimum cardinality of a total dominating set of G is the total domination number of G. Two vertices of G are said to be dotted (identified) if they are combined to form one vertex whose open neighborhood is the union of their neighborhoods minus themselves. We note that dotting any pair of vertices cannot increase the total domination number. Further we show it can decrease the total domination number by at most 2. A graph is total domination dot-stable if dotting any pair of adjacent vertices leaves the total domination number unchanged. We characterize the total domination dot-stable graphs and give a sharp upper bound on their total domination number. We also characterize the graphs attaining this bound.

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The queens graph Qn has the squares of then x n chessboard as its vertices; two squares are adjacent if they are in the same row, column or diagonal. A set D of squares of Qn is a dominating set for Qn if every square of Qn is either in D or adjacent to a square in D. If no two squares of a set I are adjacent then I is an independent set. Let 'J'(Qn) denote the minimum size of a dominating set of Qn and let i(Qn) denote the minimum size of an independent dominating set of Qn. The main purpose of this thesis is to determine new values for'!'( Qn). We begin by discussing the most important known lower bounds for 'J'(Qn) in Chapter 2. In Chapter 3 we state the hitherto known values of 'J'(Qn) and explain how they were determined. We briefly explain how to obtain all non-isomorphic minimum dominating sets for Q8 (listed in Appendix A). It is often useful to study these small dominating sets to look for patterns and possible generalisations. In Chapter 4 we determine new values for')' ( Q69 ) , ')' ( Q77 ), ')' ( Q30 ) and i (Q45 ) by considering asymmetric and symmetric dominating sets for the case n = 4k + 1 and in Chapter 5 we search for dominating sets for the case n = 4k + 3, thus determining the values of 'I' ( Q19) and 'I' (Q31 ). In Chapter 6 we prove the upper bound')' (Qn) :s; 1 8 5n + 0 (1), which is better than known bounds in the literature and in Chapter 7 we consider dominating sets on hexagonal boards. Finally, in Chapter 8 we determine the irredundance number for the hexagonal boards H5 and H7, as well as for Q5 and Q6 / Mathematical Sciences / D.Phil. (Applied Mathematics)

The Queen’s domination problem has a long and rich history. The problem can be simply stated as: What is the minimum number of queens that can be placed on a chessboard so that all squares are attacked or occupied by a queen? The problem has been expanded to include not only the standard 8x8 board, but any rectangular m×n sized board. In this thesis, we consider both paired and total domination versions of this renowned problem.

Let G = (V,E) be a graph. A set R is a restrained dominating set (total restrained dominating set, resp.) if every vertex in V − R (V) is adjacent to a vertex in R and (every vertex in V −R) to a vertex in V −R. The restrained domination number of G (total restrained domination number of G), denoted by gamma_r(G) (gamma_tr(G)), is the smallest cardinality of a restrained dominating set (total restrained dominating set) of G. If T is a tree of order n, then gamma_r(T) is greater than or equal to (n+2)/3. We show that gamma_tr(T) is greater than or equal to (n+2)/2. Moreover, we show that if n is congruent to 0 mod 4, then gamma_tr(T) is greater than or equal to (n+2)/2 + 1. We then constructively characterize the extremal trees achieving these lower bounds. Finally, if G is a graph of order n greater than or equal to 2, such that both G and G' are not isomorphic to P_3, then gamma_r(G) + gamma_r(G') is greater than or equal to 4 and less than or equal to n +2. We provide a similar result for total restrained domination and characterize the extremal graphs G of order n achieving these bounds.

The queens graph Qn has the squares of then x n chessboard as its vertices; two squares are adjacent if they are in the same row, column or diagonal. A set D of squares of Qn is a dominating set for Qn if every square of Qn is either in D or adjacent to a square in D. If no two squares of a set I are adjacent then I is an independent set. Let 'J'(Qn) denote the minimum size of a dominating set of Qn and let i(Qn) denote the minimum size of an independent dominating set of Qn. The main purpose of this thesis is to determine new values for'!'( Qn). We begin by discussing the most important known lower bounds for 'J'(Qn) in Chapter 2. In Chapter 3 we state the hitherto known values of 'J'(Qn) and explain how they were determined. We briefly explain how to obtain all non-isomorphic minimum dominating sets for Q8 (listed in Appendix A). It is often useful to study these small dominating sets to look for patterns and possible generalisations. In Chapter 4 we determine new values for')' ( Q69 ) , ')' ( Q77 ), ')' ( Q30 ) and i (Q45 ) by considering asymmetric and symmetric dominating sets for the case n = 4k + 1 and in Chapter 5 we search for dominating sets for the case n = 4k + 3, thus determining the values of 'I' ( Q19) and 'I' (Q31 ). In Chapter 6 we prove the upper bound')' (Qn) :s; 1 8 5n + 0 (1), which is better than known bounds in the literature and in Chapter 7 we consider dominating sets on hexagonal boards. Finally, in Chapter 8 we determine the irredundance number for the hexagonal boards H5 and H7, as well as for Q5 and Q6 / Mathematical Sciences / D.Phil. (Applied Mathematics)

We study the concept of strong equality of domination parameters. Let P1 and P2 be properties of vertex subsets of a graph, and assume that every subset of V(G) with property P2 also has property P1. Let ψ1(G) and ψ2(G), respectively, denote the minimum cardinalities of sets with properties P1 and P2, respectively. Then ψ1(G) ≤ ψ2(G). If ψ1(G)=ψ2(G) and every ψ1(G)-set is also a ψ2(G)-set, then we say ψ1(G) strongly equals ψ2(G), written ψ1(G) = ψ2(G). We provide a constructive characterization of the trees T such that γ(T) = i(T), where γ(T) and i(T) are the domination and independent domination numbers, respectively. A constructive characterization of the trees T for which γ(T) = γt(T), where γt(T) denotes the total domination number of T, is also presented.

A graph G is domination bicritical if the removal of any pair of vertices decreases the domination number. Properties of bicritical graphs are studied. We show that a connected bicritical graph has domination number at least 3, minimum degree at least 3, and edge-connectivity at least 2. Ways of constructing a bicritical graph from smaller bicritical graphs are presented.