The positive semidefinite zero forcing number of a graph is a graph parameter that arises from a non-traditional type of graph colouring and is related to a more conventional version of zero forcing. We establish a relation between the zero forcing and the fast–mixed searching, which implies some NP-completeness results for the zero forcing problem. Relationships between positive semidefinite zero forcing sets and clique coverings are well-understood for chordal graphs. Building upon constructions associated with optimal tree covers and forest covers, we present a linear time algorithm for computing the positive semidefinite zero forcing number of chordal graphs. We also prove that it is NP-complete to determine whether a graph has a positive semidefinite zero forcing set with an additional property.