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 total domination number γt(G) is the minimum cardinality of a total dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we investigate relationships between the annihilation number and the total domination number of a graph. Let T be a tree of order n<2. We show that γt(T)≤a(T)+1, and we characterize the extremal trees achieving equality in this bound.

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 total domination number γt(G) is the minimum cardinality of a total dominating set in G. The annihilation number a(G) is the largest integer k such that the sum of the first k terms of the non-decreasing degree sequence of G is at most the number of edges in G. In this paper, we investigate relationships between the annihilation number and the total domination number of a graph. Let T be a tree of order n<2. We show that γt(T)≤a(T)+1, and we characterize the extremal trees achieving equality in this bound.

A set S of vertices in a graph G is a total dominating set of G if every vertex is adjacent to a vertex in S. The total domination number yγ t (G) is the minimum cardinality of a total dominating set of G. The total domination subdivision number sdγt (G) of a graph G is the minimum number of edges that must be subdivided (where each edge in G can be subdivided at most once) in order to increase the total domination number. Haynes et al. (J. Combin. Math. Combin. Comput. 44 (2003) 115) showed that for any tree T of order at least 3, 1 ≤sdγt (T)≤3. In this paper, we give a constructive characterization of trees whose total domination subdivision number is 3.

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.

The domination subdivision number of a graph is the minimum number of edges that must be subdivided in order to increase the domination number of the graph. Likewise, the total domination subdivision number is the minimum number of edges that must be subdivided in order to increase the total domination number. First, this thesis provides a complete survey of established bounds on the domination subdivision number and the total domination subdivision number. Then in Chapter 4, new results regarding bounds on the total domination subdivision number are given. Finally, a characterization of the total domination subdivision number of caterpillars is presented in Chapter 5.