An approximation algorithm for zero forcing

We give an algorithm that finds a zero forcing set which approximates the optimal size by a factor of pw(G)+1, where pw(G) is the pathwidth of G. Starting from a path decomposition, the algorithm runs in $O(nm)$ time, where n and m are the order and size of the graph, respectively. As a corollary, we obtain a new upper bound on the zero forcing number in terms of the fort number and the pathwidth. The algorithm is based on a correspondence between zero forcing sets and forcing arc sets. This correspondence leads to a new bound on the zero forcing number in terms of vertex cuts, and to new, short proofs for known bounds on the zero forcing number.