AI 中文总结
本研究针对图的零强迫数的互补参数mz(G)，证明其有界时极小禁用图数量有限，确定k=3时的完整极小禁用图集，为零强迫类参数的结构理解提供新方向。
AI 中文摘要
图G的零强迫数（zero forcing number）Z(G)是一个被广泛研究的参数，源于颜色变换过程，与最小秩（minimum rank）、临界理想（critical ideals）及相关不变量有紧密联系。本研究考虑互补参数mz(G) = |V(G)| - Z(G)，该参数在取诱导子图时具有单调性，由此我们通过禁用诱导子图研究mz(G)有界的图。我们证明，对任意k≥1，满足mz(G)≤k的图的极小禁用图数量是有限的；确定了k=3时的完整极小禁用图集，并基于围长（girth）给出了mz(G)≤3的图的部分刻画。研究结果为零强迫类参数的结构理解提供了新方向。
英文摘要
The {\it zero forcing number} of a graph $G$, $Z(G)$, is a well-studied parameter which arises from a color changing process and has strong connections to {\it minimum rank}, {\it critical ideals} and related invariants. In this work, we consider the complementary parameter $\mz(G) = |V(G)| - Z(G)$. This parameter is monotone under taking induced subgraphs. This leads us to the study of graphs for which $\mz(G)$ is bounded, via forbidden induced subgraphs. We prove that the number of minimal forbidden graphs for graphs with $\mz(G)\leq k$ is finite for any $k\geq 1$. We determine the complete set of minimal forbidden graphs for the case $k = 3$, and we provide partial characterizations of graphs with $\mz(G) \leq 3$, based on girth. Our results suggest new directions for the structural understanding of zero forcing-type parameters.
