  Zero forcing is a dynamic graph coloring process whereby a colored vertex
with a single uncolored neighbor forces that neighbor to be colored. This
forcing process has been used to approximate certain linear algebraic
parameters, as well as to model the spread of diseases and information in
social networks. In this paper, we introduce and study the connected forcing
process -- a restriction of zero forcing in which the initially colored set of
vertices induces a connected subgraph. We show that the connected forcing
number -- the cardinality of the smallest initially colored vertex set which
forces the entire graph to be colored -- is a sharp upper bound to the maximum
nullity, path cover number, and leaf number of the graph. We also give closed
formulas and bounds for the connected forcing numbers of several families of
graphs including trees, hypercubes, and flower snarks, and characterize graphs
with extremal connected forcing numbers.
