Positive semidefinite zero forcing numbers of two classes of graphs

The positive semidefinite zero forcing number is a variant of the zero forcing number, which is an important parameter in the study of minimum rank/maximum nullity problems. In this paper, we consider two classes of graphs: matching-chain graphs and claw-free graphs. We first introduce the propagation decomposition of graphs; then we use this decomposition to prove a lower bound for the positive semidefinite zero forcing number of a graph. We apply this lower bound to find the positive semidefinite zero forcing number of matching-chain graphs. We show that the positive semidefinite zero forcing number and the zero forcing number agree for matching-chain graphs. As a consequence, we prove the conjecture about the positive semidefinite zero forcing number of the Cartesian product of two paths, and partially prove the conjecture about the positive semidefinite zero forcing number of the Cartesian product of a cycle and a path. For claw-free graphs, we establish a relationship between the positive semidefinite zero forcing number and the zero forcing number. We also prove that it is NP-complete to find the positive semidefinite zero forcing number of line graphs.