Yu, Bo2024-08-152024-08-1519921992https://hdl.handle.net/1828/20247A total dominating function (TDF) of a graph is a function from its vertex set to the unit interval such that the sum of function values, taken over the open neighbourhood of each vertex, is at least one. T he thesis studies some basic properties of minimal total dominating functions (MTDFs) of graphs and in particular the question of when convex combinations of MTDFs are themselves MTDFs, especially on trees. We give a necessary and sufficient condition for graphs to have a unique MTDF and various conditions for gen­eral graphs and trees to have a universal MTDF. We characterize universal MTDFs of a certain class of trees. Several classes of trees with a universal MTDF or without a universal MTDF are given.73 pagesAvailable to the World Wide WebThesis