AI 中文总结
该研究确定了路径与路径、路径与循环、循环与循环的笛卡尔积图的Grundy全支配数与斜零强制数，填补了相关界的空白并给出匹配的上下界。
AI 中文摘要
我们确定了所有两个路径或循环的笛卡尔积图的Grundy全支配数与斜零强制数，从而填补了此前针对矩形、圆柱和环面图族的非匹配界空白。对于2≤a≤b，斜零强制数Z₋(Pₐ□Pᵦ)=a−1_{{a为奇数，b为偶数}}；对于p≥2且c≥3，Z₋(Pₚ□C_c)为：当c为奇数时取min{p,c}，当c为偶数时取min{2p,c}；最后，对于3≤a≤b，Z₋(Cₐ□C_b)按不同情况取值：当a=b为奇数时取2a−1，当a=b为偶数时取2a，当a<b且a为奇数时取min{b,2a}，当a<b且a为偶数、b为奇数时取a，当a<b且a、b均为偶数时取2a。所有情形下，Grundy全支配数γᵍʳᵗ(G)=顶点数|V(G)|−Z₋(G)。含路径的下界采用斜对称或空心对称因子矩阵的克罗内克差推导，同时确定了最大斜零度与最小斜秩，还得到了最大斜零度严格小于斜零强制数的无限图族。新的循环-循环下界采用循环列缺陷估计：当较短循环为奇数时应用于完整列，当两个循环均为偶数时经单侧二分约简后应用于半列；全程通过显式强制构造给出匹配上界。
英文摘要
We determine the Grundy total domination number and the skew zero forcing number for every Cartesian product of two paths or cycles, thereby closing the nonmatching bounds previously known for these rectangular, cylindrical, and toroidal families. For $2\le a\le b$, \[ Z_-(P_a\square P_b) =a-\mathbf1_{\{a\ {\rm odd},\,b\ {\rm even}\}}. \] For $p\ge2$ and $c\ge3$, \[ Z_-(P_p\square C_c)= \begin{cases} \min\{p,c\},&c\text{ is odd}, \min\{2p,c\},&c\text{ is even}. \end{cases} \] Finally, for $3\le a\le b$, \[ Z_-(C_a\square C_b)= \begin{cases} 2a-1,&a=b\text{ odd}, 2a,&a=b\text{ even}, \min\{b,2a\},&a<b,\ a\text{ odd}, a,&a<b,\ a\text{ even},\ b\text{ odd}, 2a,&a<b,\ a,b\text{ even}. \end{cases} \] In every case, $γ_{\mathrm{gr}}^t(G)=|V(G)|-Z_-(G)$. The path-containing lower bounds use Kronecker differences of skew-symmetric or hollow symmetric factor matrices, and they also determine maximum skew nullity and minimum skew rank. They further produce an infinite family for which maximum skew nullity is strictly smaller than skew zero forcing. The new cycle--cycle lower bounds use a cyclic column-defect estimate, applied to complete columns when the shorter cycle is odd and, after a one-sided bipartite reduction, to half-columns when both cycles are even. Explicit forcing constructions give matching upper bounds throughout.
