Zero forcing sets, constrained matchings and minimum rank

The zero forcing number and the concept of constrained matching were both introduced in order to study the minimum rank of a graph. Although they have the same origin, the problem of computing the zero forcing number of a loop graph, which is a graph allowing loops, and that of computing a maximum constrained matching in a bipartite graph have been developed independently. In this paper, we highlight the equivalence of these two problems. From this result, we deduce that computing the zero forcing number of any loop graph is a NP-hard problem. Moreover, although there are algorithms computing the minimum rank of any symmetric directed tree, an efficient algorithm computing the minimum rank of any directed tree is still needed. Thanks to the equivalence between the zero forcing number and the constrained matchings, we prove that the minimum rank of a particular type of directed trees, called the loop oriented trees, can be computed in linear time.