  Zero forcing is an iterative graph coloring process whereby a colored vertex
with a single uncolored neighbor forces that neighbor to be colored. It is
NP-hard to find a minimum zero forcing set - a smallest set of initially
colored vertices which forces the entire graph to be colored. We show that the
problem remains NP-hard when the initially colored set induces a connected
subgraph. We also give structural results about the connected zero forcing sets
of a graph related to the graph's density, separating sets, and certain induced
subgraphs, and we characterize the cardinality of the minimum connected zero
forcing sets of unicyclic graphs and variants of cactus and block graphs.
Finally, we identify several families of graphs whose connected zero forcing
sets define greedoids and matroids.
