The $Z_q$-forcing number for some graph families

The zero forcing number was introduced as a combinatorial bound on the maximum nullity taken over the set of real symmetric matrices that respect the pattern of an underlying graph. The $Z_q$-forcing game is an analog to the standard zero forcing game which incorporates inertia restrictions on the set of matrices associated with a graph. This work proves an upper bound on the $Z_q$-forcing number for trees. Furthermore, we consider the $Z_q$-forcing number for caterpillar cycles on $n$ vertices. We focus on developing game theoretic proofs of upper and lower bounds.