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

TL;DR

This paper provides a comprehensive hierarchy analysis of directed and undirected temporal graphs based on support, reachability, and equivalence notions, revealing fundamental differences and transformations between classes.

Contribution

It extends existing frameworks by adding directed vs. undirected distinctions and characterizes the hierarchy and transformations of temporal graph classes.

Findings

Directed graphs form a single hierarchy with strict & simple as most expressive.

Undirected graphs form a two-strand hierarchy with distinct expressive classes.

Reachability equivalence allows transforming undirected classes into directed ones, but not vice versa.

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.