A New Lower Bound for Positive Zero Forcing
The positive 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 first introduce the propagation decomposition of graphs; then we use this decomposition to prove a lower bound for the positive zero forcing number of a graph. We apply this lower bound to find the positive zero forcing number of matching-chain graphs. We prove that the positive zero forcing number of a matching-chain graph is equal to its zero forcing number. As a consequence, we prove the conjecture about the positive zero forcing number of the Cartesian product of two paths, and partially prove the conjecture about the positive zero forcing number of the Cartesian product of a cycle and a path. We also show that the positive zero forcing number and the zero forcing number agree for claw-free graphs. We prove that it is NP-complete to find the positive zero forcing number of line graphs.
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