Simple, strict, proper, happy: A study of reachability in temporal graphs

TL;DR

This paper analyzes how three subtleties—strictness of temporal paths, properness of time labels, and simpleness of edges—shape reachability in temporal graphs through the reachability graph abstraction. It establishes a near-complete hierarchy of expressivity among six meaningful settings, identifies key separations, and presents three transformations (dilation, saturation, semaphore) that relate settings and preserve reachability properties. The authors advocate happy graphs (proper and simple) as a minimal yet informative model, showing that negative results in this setting carry over to broader settings and successfully extending known hardness and non-existence results (temporal components and sparse spanners) to happy graphs. The work provides a principled guide for selecting temporal-graph models in research and highlights open questions about realizability, characterization, and directed extensions.

Abstract

Dynamic networks are a complex subject. Not only do they inherit the complexity of static networks (as a particular case); they are also sensitive to definitional subtleties that are a frequent source of confusion and incomparability of results in the literature. In this paper, we take a step back and examine three such aspects in more details, exploring their impact in a systematic way; namely, whether the temporal paths are required to be \emph{strict} (i.e., the times along a path must increasing, not just be non-decreasing), whether the time labeling is \emph{proper} (two adjacent edges cannot be present at the same time) and whether the time labeling is \emph{simple} (an edge can have only one presence time). In particular, we investigate how different combinations of these features impact the expressivity of the graph in terms of reachability. Our results imply a hierarchy of expressivity for the resulting settings, shedding light on the loss of generality that one is making when considering either combination. Some settings are more general than expected; in particular, proper temporal graphs turn out to be as expressive as general temporal graphs where non-strict paths are allowed. Also, we show that the simplest setting, that of \emph{happy} temporal graphs (i.e., both proper and simple) remains expressive enough to emulate the reachability of general temporal graphs in a certain (restricted but useful) sense. Furthermore, this setting is advocated as a target of choice for proving negative results. We illustrates this by strengthening two known results to happy graphs (namely, the inexistence of sparse spanners, and the hardness of computing temporal components). Overall, we hope that this article can be seen as a guide for choosing between different settings of temporal graphs, while being aware of the way these choices affect generality.

Figures (10)

• Figure 1: Settings resulting from combining the three properties.
• Figure 2: Some temporal graphs on four vertices.
• Figure 3: Dilation of the labels of $\mathcal{G}$. Here, a single snapshot, namely $G_1$ contains paths whose length is larger than $1$, so the dilation is only applied to $G_1$. The transformed snapshot $\mathcal{G}_1$ has $6$ labels (instead of $1$). It is then recomposed with the other snapshots, whose time labels are shifted accordingly (by $5$ time unit), resulting in graph $\mathcal{H}$.
• Figure 4: The semaphore technique, turning a non-proper graph $\mathcal{G}$ (in the strict setting), into a happy graph $\mathcal{H}$ whose reachability preserves the relation among original vertices.
• Figure 5: Separations, transformations, and inclusions among settings.

Theorems & Definitions (35)

• Definition 1: Reachability equivalence
• Definition 2: Support equivalence
• Definition 3: Bijective equivalence
• Definition 4: Induced reachability equivalence
• Lemma 1
• Lemma 2
• proof
• Corollary 1
• Lemma 3
• proof