Zero forcing number of graphs

A subset $S$ of initially infected vertices of a graph $G$ is called forcing if we can infect the entire graph by iteratively applying the following process. At each step, any infected vertex which has a unique uninfected neighbour, infects this neighbour. The forcing number of $G$ is the minimum cardinality of a forcing set in $G$. In the present paper, we study the forcing number of various classes of graphs, including graphs of large girth, $H$-free graphs for a fixed bipartite graph $H$, random and pseudorandom graphs.