A Two-Player Zero Forcing Game

Abstract

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.