AI 中文总结
该研究确定了矩形网格$P_m\square P_n$上3邻域自举渗流的最小渗流集大小，扩展了前人工作，还针对环面网格$C_m\square C_n$得到了相差不超过1的上下界及部分情形的精确解。
AI 中文摘要
在3邻域自举渗流过程中，若一个顶点的至少三个邻居被感染，则该顶点会被感染并保持感染状态。若初始感染顶点的构型最终使所有顶点都被感染，则称其发生渗流。我们精确确定了矩形网格图$P_m\square P_n$上3邻域自举渗流过程所有剩余开放情形的最小渗流集大小，这扩展了Dukes、Noel和Romer的早期工作。此外，我们针对环面网格$C_m\square C_n$研究相同问题，证明了上下界相差不超过1，并在许多可除情形下精确确定了答案。
英文摘要
In the $3$-neighbor bootstrap percolation process, a vertex becomes (and remains) infected if at least three of its neighbors are infected. We say that an initial configuration of infected vertices percolates if eventually all vertices are infected. We exactly determine the size of the minimum percolating set for the $3$-neighbor bootstrap percolation process on all remaining open cases for rectangular grid graphs $P_m\square P_n$. This extends earlier work of Dukes, Noel, and Romer. Additionally, we consider the same question for the toroidal grids $C_m\square C_n$, proving upper and lower bounds which are at most one apart and determining the answer precisely in many divisibility cases.
Comments39 pages, 23 figures.x
