Exploring Temporal Graphs with Frequent and Regular Edges

TL;DR

The paper studies exploration in temporal graphs under two activation paradigms: frequent edges with frequency $f_e$ and regular edges with regularity $r_e$. It introduces a frequency-weighted spanning-tree approach: build $FW(\,G)$ with weights $f_e$, compute a minimum weight spanning tree, and traverse it to obtain a temporal exploration; since a tree on $n$ vertices admits a covering walk of length at most $2n-3$, the authors derive upper bounds $f(2n-3)$ for $f$-frequent graphs and $r(2n-3)$ for $r$-regular graphs. They further instantiate these results for restricted classes, including Public Transport Graphs (bound $(2n-3) imes L_{ ext{max}}$), Sequential Connection Graphs (bound $4|E|$), and Broadcast Networks (general bound $d n(2n-3)$, with a tighter $( abla+1)(2n-3)$ for always-connected networks). The work provides both algorithmic procedures and complexity considerations, offering practical upper bounds and motivating further study of tightness and optimality in these structured temporal settings. The findings have potential impact on planning and analysis of real-world time-varying networks, such as transit systems and distributed communications, by guiding efficient exploration schedules under predictable edge activation patterns.

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.

Figures (1)

• Figure 1: Overview of the exploration of the temporal graph $\mathcal{G}$ with $\mathcal{G} = (V, E_1, E_2, E_3, E_1, E_2, E_3, \dots, E_1, E_2, E_3)$, i.e. the temporal graph formed by repeating the edge sets $E_1$, $E_2$ and $E_3$ some number of times.

Theorems & Definitions (17)

• Lemma 3.1
• Corollary 3.2
• Lemma 3.3: kruskal1956shortest
• Lemma 3.4: Folklore
• Theorem 3.5
• Lemma 3.6
• Corollary 3.7
• Corollary 3.8
• Corollary 3.9
• Theorem 3.10