Living Temporal Graphs II: Snapshots, Movies, and Temporal Homomorphisms

This record contains the preprint version of the manuscript entitled “Living Temporal Graphs II: Snapshots, Movies, and Temporal Homomorphisms”. In the first part of this work, we introduced a foundational framework for living temporal graphs with stateful vertices — Active, Sleep, DeadIn, DeadOut, and DeadBoth — together with two classes of temporal edges: continuous-flow edges, which require simultaneous endpoint activity, and discrete-transport edges, which require only admissible departure and arrival. In this sequel, we develop the structural theory of living temporal graphs. The paper focuses on three interrelated themes: snapshots and movies as ways to observe a living temporal graph at an instant or over an interval; temporal homomorphisms as tools for comparing and reducing such graphs; and a local temporal geometry based on critical times, hills, holes, and saddles. We prove reconstruction theorems, establish the categorical structure of the resulting category, and present five algorithmic procedures with complexity analysis. Applications to video analysis, sensor networks, transportation planning, and machine learning on temporal graphs are also discussed. The manuscript concludes with six open conjectures for future research.