2025/09/25 by Ryan Moruzzi, Moruzzi, Ryan, Shah, Sagar +2
Computer Science · Engineering · #05C50 #05C76 #Advanced Graph Theory Research #Combinatorics (math.CO) #Computational Geometry and Mesh Generation #FOS: Mathematics #Structural Analysis and Optimization
paper · pdf · doi:10.48550/arxiv.2509.21529
openalex publication_date 2025/09/25 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/29
We study zero forcing and ℓ-leaky zero forcing on induced subgraphs of d-dimensional grid graphs. Using ℓ-leaky forts, we prove structural results showing that for ℓ ≤ 2d-1, every nonempty ℓ-leaky fort in an induced subgraph of Pn1\square⋯\square Pnd intersects the boundary of the graph. These results give general bounds and, in certain settings, exact values for the ℓ-leaky forcing number of induced subgraphs. Motivated by this framework, we introduce an integer lattice based definition of the Hopi rectangle graphs HD(a,b) as induced subgraphs of Pa+b\square Pa+b. For this particular family of graphs, we show that the zero forcing number equals the maximum nullity, and we completely characterize the ℓ-leaky forcing number for all ℓ≥ 1.