On Sufficient Conditions for Short Journeys in Temporal Graphs

Abstract

A temporal graph is defined as a sequence (G₁, G₂, …, G_L) of static graphs on a common set of n vertices. A strict journey in a temporal graph is the temporal analogue of a path in a static graph, in which at most one edge may be traversed at each time step.
There exists a notable connection between the existence of paths in static graphs and the existence of strict journeys in specific temporal graphs. A well-known folklore result, commonly referred to as the Reachability Lemma, states that for two vertices u and v, if there are at least n-1 time steps during which a path connects u and v, then a strict journey from u to v exists.
Our main theorem extends this lemma. Under the same assumptions as those of the Reachability Lemma, we prove that a strict journey from u to v exists and the number of edges traversed by such a journey admits a non-trivial upper bound. Furthermore, this bound converges toward the average length of the paths connecting u and v as the number of such paths increases. A corresponding lower bound is also established.
In the second part of this work, we investigate the setting in which every path connecting vertices u and v has length at most a given integer k. For an integer b ≥ k, we characterize the sufficient number of time steps containing such a path that guarantees the existence of a journey from u to v traversing at most b edges. We derive an upper bound of ⌊(n-k-1)/(b-k+1) ⋅ (b-1)⌋ + k, and a lower bound of ⌊(n-k-1)/(b -k+1)⌋ ⋅ (b-1) + r + k-1, where r = (n-k-1 mod (b-k+1)). Finally, we present several applications of the first theorem, with particular emphasis on always connected temporal graphs, that is, temporal graphs where at each time step the graph is connected.