Simple, Strict, Proper, and Directed: Comparing Reachability in Directed and Undirected Temporal Graphs

TL;DR

This work extends the temporal-graph framework of Casteigts et al. to directed graphs, analyzing how directionality interacts with strict/non-strict timing, proper/arbitrary adjacency, and simple/multi-labeled edges under bijective, support, reachability, and induced-reachability equivalences. The authors develop directed analogues of dilation, saturation, and semaphore transformations, proving a single-strand, reachability-based hierarchy for directed settings and showing that all directed settings are induced-reachability equivalent. They complete the undirected hierarchy from prior work, establishing a two-strand structure and resolving open questions, and then compare the two to obtain a merged view: directed classes are strictly more expressive than undirected ones, and reachability can be preserved from UD to D, but not vice versa. The results enable transferring positive/negative complexity and algorithmic insights across settings and motivate a unified approach to temporal graphs under different modeling choices.

Abstract

We present the first comprehensive analysis of temporal settings for directed temporal graphs, fully resolving their hierarchy with respect to support, reachability, and induced-reachability equivalence. These notions, introduced by Casteigts, Corsini, and Sarkar, capture different levels of equivalence between temporal graph classes. Their analysis focused on undirected graphs under three dimensions: strict vs. non-strict (whether times along paths strictly increase), proper vs. arbitrary (whether adjacent edges can appear simultaneously), and simple vs. multi-labeled (whether an edge can appear multiple times). In this work, we extend their framework by adding the fundamental distinction of directed vs. undirected. Our results reveal a single-strand hierarchy for directed graphs, with strict & simple being the most expressive class and proper & simple the least expressive. In contrast, undirected graphs form a two-strand hierarchy, with strict & multi-labeled being the most expressive and proper & simple the least expressive. The two strands are formed by the non-strict & simple and the strict & simple class, which we show to be incomparable. In addition to examining the internal hierarchies of directed and of undirected graph classes, we compare the two. We show that each undirected class can be transformed into its directed counterpart under reachability equivalence, while no directed class can be transformed into any undirected one. Our findings have significant implications for the study of computational problems on temporal graphs. Positive results in more expressive graph classes extend to weaker classes as long as the problem is independent under reachability equivalence. Conversely, hardness results for a less expressive class propagate to stronger classes. We hope these findings will inspire a unified approach for analyzing temporal graphs under the different settings.

Figures (19)

• Figure 1: Illustration of the reachability hierarchy for the temporal graph settings. An arrow from (1) to (2) indicates that every graph in (1) can be transformed into a graph in (2) with the same reachability graph. Indirect transformations are implied. The two red dotted arrows represent the open questions. The orange boxes (top) show the directed hierarchy (\ref{['sec:directed-reachability']}) and the blue boxes (bottom) the undirected hierarchy (\ref{['sec:undirected-reachability']}). The merged hierarchy is discussed in \ref{['sec:directed-VS-undirected']}.
• Figure 2: On the left is an illustration of the subset relations between the settings. An arrow from (1) to (2) represents that (1) is a subset of (2). This holds for both directed and undirected graphs. On the right is an illustration adopted from CCS, showing the subset relation in a venn diagram.
• Figure 3: The reachability hierarchy of the undirected graph settings. The undirected hierarchy under support equivalence is the same. With respect to induced-reachability equivalence, all undirected settings are equivalent.
• Figure 4: The reachability hierarchy of the directed graph settings. The directed hierarchy under support equivalence is different; it is the same hierarchy as in \ref{['fig:hierarchy-undirected']}. With respect to induced-reachability equivalence, all directed settings are equivalent.
• Figure 5: Directed triangle with time label 1 on each edge (left) and its reachability graph in the D & strict (& simple) setting (right).

Theorems & Definitions (35)

• definition 1: Bijective equivalence
• definition 2: Support equivalence
• definition 3: Reachability equivalence
• definition 4: induced-Reachability equivalence
• theorem 1: UD & non-strict & simple $\not\rightsquigarrow^{R}$ UD & strict & simple
• proof
• lemma 1
• proof
• lemma 2
• proof