  A continuous-time quantum walk on a dynamic graph evolves by Schr\"odinger's
equation with a sequence of Hamiltonians encoding the edges of the graph. This
process is universal for quantum computing, but in general, the dynamic graph
that implements a quantum circuit can be quite complicated. In this paper, we
give six scenarios under which a dynamic graph can be simplified, and they
exploit commuting graphs, identical graphs, perfect state transfer,
complementary graphs, isolated vertices, and uniform mixing on the hypercube.
As examples, we simplify dynamic graphs, in some instances allowing
single-qubit gates to be implemented in parallel.
