A graph G with no isolated vertex is total domination vertex critical if for any vertex v of G that is not adjacent to a vertex of degree one, the total domination number of G - v is less than the total domination number of G. These graphs we call γt-critical. If such a graph G has total domination number k, we call it k-γt-critical. We characterize the connected graphs with minimum degree one that are γ t-critical and we obtain sharp bounds on their maximum diameter. We calculate the maximum diameter of a k-γt-critical graph for k≤8 and provide an example which shows that the maximum diameter is in general at least 5k/3 - O(1).

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.

A set S of vertices of a graph G = (V, E) is a dominating set if every vertex in V - S is adjacent to some vertex in 3. The domination number γ(G) is the minimum cardinality of a dominating set of G, and the domination subdivision number sdγ(G) is the minimum number of edges that must be subdivided (each edge in G can be subdivided at most once) in order to increase the domination number. In June 2000, Arumugam conjectured that 1 ≤ sdγ(G) ≤ 3 for any graph G. However, a counterexample to this conjecture given in [6] suggests the modified conjecture that 1 ≤ sdγ(G) ≤ 4 for any graph G. It is also conjectured in [6] that for every graph G with minimum degree δ(G) ≥ 2, sdγ(G) ≤ δ(G) + 1. In this paper we extend several previous results and consider evidence in support of these two conjectures.

The lower and upper independent, domination and irredundant numbers of the graph G = (V, E) are denoted by i ( G) , f3 ( G), 'Y ( G), r ( G), ir ( G) and IR ( G) respectively. These six numbers are called the domination parameters. For each of these parameters n:, we define six types of criticality. The graph G is n:-critical (n:+ -critical) if the removal of any vertex of G causes n: (G) to decrease (increase), G is n:-edge-critical (n:+-edge-critical) if the addition of any missing edge causes n: (G) to decrease (increase), and G is Ir-ER-critical (n:- -ER-critical) if the removal of any edge causes n: (G) to increase (decrease). For all the above-mentioned parameters n: there exist graphs which are n:-critical, n:-edge-critical and n:-ER-critical. However, there do not exist any n:+-critical graphs for n: E {ir,"f,i,/3,IR}, no n:+-edge-critical graphs for n: E {ir,"f,i,/3} and non:--ER-critical graphs for: E {'Y,/3,r,IR}. Graphs which are "I-critical, i-critical, "I-edge-critical and i-edge-critical are well studied in the literature. In this thesis we explore the remaining types of criticality. We commence with the determination of the domination parameters of some wellknown classes of graphs. Each class of graphs we consider will turn out to contain a subclass consisting of graphs that are critical according to one or more of the definitions above. We present characterisations of "I-critical, i-critical, "I-edge-critical and i-edge-critical graphs, as well as ofn:-ER-critical graphs for n: E {/3,r,IR}. These characterisations are useful in deciding which graphs in a specific class are critical. Our main results concern n:-critical and n:-edge-critical graphs for n: E {/3, r, IR}. We show that the only /3-critical graphs are the edgeless graphs and that a graph is IRcritical if and only if it is r-critical, and proceed to investigate the r-critical graphs which are not /3-critical. We characterise /3-edge-critical and r-edge-critical graphs and show that the classes of IR-edge-critical and r-edge-critical graphs coincide. We also exhibit classes of r+ -critical, r+ -edge-critical and i- -ER-critical graphs. / Mathematical Sciences / D. Phil. (Mathematics)

The lower and upper independent, domination and irredundant numbers of the graph G = (V, E) are denoted by i ( G) , f3 ( G), 'Y ( G), r ( G), ir ( G) and IR ( G) respectively. These six numbers are called the domination parameters. For each of these parameters n:, we define six types of criticality. The graph G is n:-critical (n:+ -critical) if the removal of any vertex of G causes n: (G) to decrease (increase), G is n:-edge-critical (n:+-edge-critical) if the addition of any missing edge causes n: (G) to decrease (increase), and G is Ir-ER-critical (n:- -ER-critical) if the removal of any edge causes n: (G) to increase (decrease). For all the above-mentioned parameters n: there exist graphs which are n:-critical, n:-edge-critical and n:-ER-critical. However, there do not exist any n:+-critical graphs for n: E {ir,"f,i,/3,IR}, no n:+-edge-critical graphs for n: E {ir,"f,i,/3} and non:--ER-critical graphs for: E {'Y,/3,r,IR}. Graphs which are "I-critical, i-critical, "I-edge-critical and i-edge-critical are well studied in the literature. In this thesis we explore the remaining types of criticality. We commence with the determination of the domination parameters of some wellknown classes of graphs. Each class of graphs we consider will turn out to contain a subclass consisting of graphs that are critical according to one or more of the definitions above. We present characterisations of "I-critical, i-critical, "I-edge-critical and i-edge-critical graphs, as well as ofn:-ER-critical graphs for n: E {/3,r,IR}. These characterisations are useful in deciding which graphs in a specific class are critical. Our main results concern n:-critical and n:-edge-critical graphs for n: E {/3, r, IR}. We show that the only /3-critical graphs are the edgeless graphs and that a graph is IRcritical if and only if it is r-critical, and proceed to investigate the r-critical graphs which are not /3-critical. We characterise /3-edge-critical and r-edge-critical graphs and show that the classes of IR-edge-critical and r-edge-critical graphs coincide. We also exhibit classes of r+ -critical, r+ -edge-critical and i- -ER-critical graphs. / Mathematical Sciences / D. Phil. (Mathematics)