This thesis exhibits a collection of combinatorial optimization problems and the integer programs proposed to solve them based on new mathematical insights. In particular, graph propagation and graph throttling problems including the positive semidefinite zero forcing set problem and the minimum power dominating set problem are considered, as well as the graph connectivity problem known as the strong rainbow connection problem. A parallel treatment of the graph propagation problems is provided in which set cover problems are defined using problem specific blocking sets. These blocking sets are introduced, their structural properties are investigated, and computational methods for identifying them are proposed, providing a general recipe for developing integer programming approaches for graph propagation problems. The strong rainbow connection problem is also studied, and the first general computational method for the problem is introduced. New lower bounds, computational enhancements, and an alternative solution method based on iterative lower bound improvement are also proposed, the latter of which is shown to be highly effective in practice.