Zero Forcing, Support Varieties, and Boundary Complexes

Source abstract

Let a finite simple graph be given. We study symmetric matrices with this graph pattern that admit a null vector with prescribed exact support. Such a vector exists precisely when its support is a fort, meaning that no vertex outside the support has exactly one neighbor inside it. Over the complex numbers, we prove that the Zariski closure of each exact-support coefficient locus is irreducible and compute its codimension in terms of boundary vertices and a labeled forest on the connected components of the support. The corresponding forest deficiency also determines the generic restricted nullity. An incidence parametrization yields a prime decomposition of the reduced zero-forcing failure locus. For a prescribed common kernel, we describe the coefficient space through equilibrium stresses, exterior affine dependencies, and free coefficients. We then construct boundary complexes recording local solution values and equation residuals. Their gluing maps are defined over the universal coefficient ring and commute with arbitrary base change. By retaining the complete collection of support signatures associated with each seed set, we obtain a composition formula for the minimum zero-forcing number and an algorithm for its computation.