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Characterizations of the Connected Forcing Number of a Graph

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.