Grundy Total Domination and Skew Zero Forcing in Cartesian Products of Paths and Cycles

Source abstract

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.