Exploring Temporal Graphs with Frequent and Regular Edges

Abstract

Temporal graphs are a class of graphs defined by a constant set of vertices and a changing set of edges, each of which is known as a timestep. These graphs are well motivated in modelling real-world networks, where connections may change over time. One such example, itself the primary motivation for this paper, are public transport networks, where vertices represent stops and edges the connections available at some given time. Exploration problems are one of the most studied problems for temporal graphs, asking if an agent starting at some given vertex $v$ can visit every vertex in the graph. In this paper, we study two primary classes of temporal graphs. First, we study temporal graphs with \emph{frequent edges}, temporal graphs where each edge $e$ is active at least once every $f_e$ timesteps, called the frequency of the edge. Second, temporal graphs with \emph{regular edges}, graphs where each edge $e$ is active at any timestep $t$ where $t \equiv s_e \bmod r_e$, with $s_e$ being the start time of the edge, and $r_e$ the regularity. We show that graphs with frequent edges can be explored in $O(F n)$ timesteps, where $F = \max_{e \in E} f_e$, and that graphs with regular edges can be explored in $O(R n)$ timesteps, where $R = \max_{e \in E} r_e$. We provide additional results for \emph{public transport graphs}, temporal graphs formed by the union of several routes, corresponding to the schedules of some modes of transit, for \emph{sequential connection graphs}, temporal graphs in which each vertex has a single active in-edge per timestep, iterating over the set of edges in some order, and for \emph{broadcast networks}, a representation of communication within distributed networks where each vertex broadcasts a message either to all vertices, or none at each timestep.