A Two-Player Zero Forcing Game

We introduce a competitive two-player zero forcing game on a connected graph. Alice and Bob alternately seed white vertices or perform legal zero forces in their own colours, and each player seeks to minimise their own number of seeds. A force preservation rule prevents avoidable blocking of an opponent's established force. Because distinct continuations can be equally good for the player to move, optimal play is defined by a set-valued backward induction, and $Z_g(G)$ is the minimum total number of seeds among the resulting optimal outcomes. We prove that $Z_g(G)\geq Z(G)$, determine $Z_g$ for paths, cycles, stars, complete graphs, and complete bipartite graphs, and characterise the graphs with $Z_g(G)=2$ by an alternating two-chain forcing schedule. We also show that $Z_g$ is not minor-monotone and that edge subdivision can either increase or decrease the parameter. Exact computation verifies $Z_g(G)\leq2Z(G)$ through order nine.