This paper is the fourth part of the Living Temporal Graphs series. It extends the previous state-sensitive framework by studying how temporal paths are formed when vertices are subject to local restrictions such as saturation, capacity, and edge-flow control. In this setting, vertices are treated as living entities with lifetimes and changing states, while directed edges are temporal objects whose birth and death depend on both time and vertex conditions. The paper focuses on how local vertex constraints affect the admissibility of edge births and deaths, and how these constraints shape different types of temporal paths. We introduce and organise several path notions, including regular, irregular, connected, disconnected, strongly regular, bounded-gap, and constraint-admissible paths. A central contribution of the paper is the study of Gap-Overlap constraints and their interaction with local saturation and capacity bounds. The paper proves several structural results, including an event-static interval criterion, global birth–death balance, distance inflation, weak saturation cut theorem, obstruction and bottleneck theorems, and a boundary theorem separating constraint-admissible behaviour from irregular path formation. The work also highlights future directions in constrained temporal reachability, reconstruction, stability, optimisation, cut duality, fixed-parameter tractability, and algorithmic questions on time-expanded structures. Overall, this paper develops a more detailed language for analysing movement, communication, and flow in living temporal graphs under local vertex-based constraints.
Gordji et al. (Tue,) studied this question.