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.

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.