This paper presents graph transformation algorithms for register-pressure-aware instruction scheduling. The proposed transformations add edges to the data dependence graph (DDG) to eliminate solutions that are either redundant or sub-optimal. Register-pressure-aware instruction scheduling aims at balancing two conflicting objectives: maximizing instruction-level parallelism (ILP) and minimizing register pressure (RP). Graph transformations have been previously proposed for the problem of maximizing ILP without considering RP, which is a problem of limited practical value. In the current paper, we extend that work by proposing graph transformations for the RP minimization objective, which is an important objective in practice. Various cost functions are considered for representing RP, and we show that the proposed transformations preserve optimality with respect to each of them. The proposed transformations are used to reduce the size of the solution space before applying a Branch-and-Bound (B&B) algorithm that exhaustively searches for an optimal solution. The proposed transformations and the B&B algorithm were implemented in the LLVM compiler, and their performance was evaluated experimentally on a CPU target and a GPU target. The SPEC CPU2017 floating-point benchmarks were used on the CPU and the PlaidML benchmarks were used on the GPU. The results show that the proposed transformations significantly reduce the compile time while giving approximately the same execution-time performance.