Graph models for covariant holographic entropy I

We construct a graph model for holographic entropies in general time-dependent spacetimes. In static settings, such models arise from Ryu-Takayanagi surfaces on a common Cauchy slice and imply that the holographic entropy cone is polyhedral. Extending this construction to the covariant Hubeny-Rangamani-Takayanagi (HRT) setting is obstructed by the absence of a preferred time slice, raising the possibility of unphysical "short-cuts" built from partial HRT surfaces. We identify a geometric condition--the existence of exposed regions for each pair of HRT surfaces--under which this obstruction is removed. Under this condition, we construct weight functions by projecting along null generators of entanglement horizons and prove a Conditional No-Short-Cut Theorem: any graph cut is dominated by a surface composed of complete HRT surfaces. Consequently, the graph model reproduces HRT entropies, establishing the equivalence between the covariant and static holographic entropy cones in this regime. We further show that configurations in which exposed regions are absent due to nesting of interaction regions can be partially resolved by grouping HRT surfaces into timelike clusters. This provides evidence that the graph model extends beyond the exposed-region regime and suggests a path toward a complete covariant construction.