On the $Z_q$-forcing number: computational approach and exact values

Zero forcing is a graph coloring process that is used to model spreading phenomena in real-world scenarios. It can also be viewed as a single-player combinatorial game on a graph, where the player's goal is to select a subset of vertices of minimum cardinality that eventually leads to all vertices of the graph being colored. A variant of this game, called the $q$-analogue of zero forcing, was later introduced. In this version, the player again seeks to choose the smallest number of vertices that will eventually color the entire graph, while an oracle attempts to force the player to select a larger subset. In this paper, we exploit the structural properties of several graph classes in order to both derive algorithms to compute the exact value of $Z_q$, and to establish bounds and exact values of the parameter for these graph classes. In particular, we present a SAT-based algorithm to compute the $q$-analogue zero forcing number, offering optimal strategies for both the player and the oracle. Additionally, we propose a polynomial-time algorithm for computing the $q$-analogue zero forcing number for $q=1$ of cactus graphs. Lastly, we prove the exact value of this parameter for several graph classes such as block graphs. Our work extends previous results about trees by Butler et al. (2020) and Blanco et al. (2024).