Forts, (fractional) zero forcing, and Cartesian products of graphs
Thomas R. Cameron, Leslie Hogben, Franklin H. J. Kenter, Seyed Ahmad Mojallal, Houston Schuerger
TL;DR
The paper develops a hypergraph-based framework for zero forcing via forts, introducing the fort hypergraph $\mathcal{F}_G$ and its transversal and matching numbers to define the fractional zero forcing number $\mathop{\mathrm{Z}}^{*}(G)$ and the fort number $\mathop{\mathrm{ft}}(G)$. It proves key dualities $\mathop{\mathrm{Z}}^{*}(G)=\mathop{\mathrm{ft}}^{*}(G)$ and the bounds $\mathop{\mathrm{ft}}(G) \le \mathop{\mathrm{Z}}^{*}(G) \le \mathop{\mathrm{Z}}(G)$, then leverages hypergraph transversal results to obtain Vizing-like lower bounds for Cartesian products: in particular, $\mathop{\mathrm{Z}}(G\Box G') \ge \mathop{\mathrm{Z}}(G)\mathop{\mathrm{Z}}^{*}(G')+1$ whenever $\mathop{\mathrm{Z}}(G)=\mathop{\mathrm{Z}}^{*}(G)$, and $\mathop{\mathrm{ft}}(G\Box G') \ge \mathop{\mathrm{ft}}(G)\mathop{\mathrm{ft}}(G')$, $\mathop{\mathrm{Z}}^{*}(G\Box G') \ge \mathop{\mathrm{Z}}^{*}(G)\mathop{\mathrm{Z}}^{*}(G')$. The authors compute exact $\mathop{\mathrm{ft}}(G)$ and $\mathop{\mathrm{Z}}^{*}(G)$ for many graph families (paths, cycles, complete graphs, complete bipartite graphs, coronas, Petersen graph, hypercubes, etc.), identify broad families with $\mathop{\mathrm{Z}}^{*}(G)=\mathop{\mathrm{Z}}(G)$ (including trees in a class $\mathcal{T}$ and several two-parallel-path structures), and present extremal constructions (notably star-clique paths) yielding equality in the product bounds. The concluding discussion connects these parameters to maximum nullity $\operatorname{M}$, motivates open questions, and provides a coherent set of targeted conjectures and directions for future study. The results deepen the connection between zero forcing, hypergraph transversals, and the algebraic question of maximum nullity, with concrete implications for bounds on $\operatorname{M}$ and the minimum ranks of Cartesian products.
Abstract
The (disjoint) fort number and fractional zero forcing number are introduced and related to existing parameters including the (standard) zero forcing number. The fort hypergraph is introduced and hypergraph results on transversals and matchings are applied to the zero forcing number and fort number. These results are used to establish a Vizing-like lower bound for the zero forcing number of a Cartesian product of graphs for certain families of graphs, and a family of graphs achieving this lower bound is exhibited.