Living Temporal Graphs III: Local Saturation, Capacity, and Vertex-Based Edge Flow Control

This record contains the manuscript entitled “Living Temporal Graphs III: Local Saturation, Capacity, and Vertex-Based Edge Flow Control”. The manuscript is the third part of the Living Temporal Graphs series. Building on the foundational framework introduced in Parts I and II, it develops a new layer of local temporal dynamics based on interval-dependent vertex constraints. These constraints regulate the admissibility of edge births and edge deaths through notions of out-saturation, in-saturation, bilateral saturation, and local capacity bounds. The work distinguishes these saturation and capacity constraints from intrinsic vertex states such as Active, Sleep, DeadIn, DeadOut, and DeadBoth. While vertex states describe the internal temporal condition of a vertex, the constraints introduced in this manuscript act as external or endogenous local controls over specified time intervals. The manuscript formalizes the mathematical structure of locally constrained living temporal graphs and establishes several structural results, including monotonicity, additivity, obstruction theorems, an event-static interval criterion, a global birth–death balance theorem, a distance inflation theorem, and a weak saturation cut theorem. It also discusses reconstruction, stability, optimization, and reachability problems arising from vertex-based edge flow control. The paper concludes with ten open conjectures and problems concerning constrained temporal diameter, computational complexity, Helly-type saturation phenomena, cut duality, reconstruction stability, fixed-parameter tractability, and polynomial solvability on time-DAGs.