  The positive 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. For chordal graphs much is understood
regarding the relationships between positive zero forcing and clique coverings.
Building upon constructions associated with optimal tree covers and forest
covers, we present a linear time algorithm for computing the positive zero
forcing number of chordal graphs. We also prove that it is NP-complete to
determine if a graph has a positive zero forcing set with an additional
property.
