Menger's Theorem for Temporal Paths (Not Walks)

TL;DR

This paper extends Menger's Theorem to directed temporal graphs, analyzing connectivity and robustness through various definitions of disjoint paths and paths' temporal properties, with a key result for the case of one disjoint path.

Contribution

It introduces a comprehensive framework for temporal paths and disjointness, and proves Menger's Theorem holds when at most one temporal vertex-disjoint path exists.

Findings

Menger's Theorem holds for the case of a single temporal vertex-disjoint path.

Different interpretations of disjointness and path strictness are systematically analyzed.

New theoretical results unify connectivity concepts in temporal graphs.

Abstract

A (directed) temporal graph is a (directed) graph whose edges are available only at specific times during its (discretized) lifetime $\tau$. In this setting, we ask that walks respect the temporal aspect by defining $\textit{temporal walks}$ as sequences of adjacent edges whose appearing times are either strictly increasing or non-decreasing (here called non-strict), depending on the scenario. The notion of disjointness between walks is also not unique: two walks are $\textit{vertex-disjoint}$ if they do not share a vertex, and are $\textit{temporal vertex-disjoint}$ if they do not share a vertex at the same time. Thus a $\textit{temporal path}$ is a temporal walk where no repetition of vertices, at any time, is allowed. This is an important distinction that separates the interpretation of our results from those of previous works on the topic. In this paper we focus on various questions regarding connectivity (maximum number of disjoint paths) and robustness (minimum size of a cut) between a given pair of vertices. Such problems are related to the well-known Menger's Theorem on static graphs. We explore all possible interpretations of such problems, according to vertex and temporal vertex-disjointness, strict and non-strict temporal paths, and directed and undirected temporal graphs. We present a number of new results, the main of which states that Menger's Theorem holds when the maximum number of temporal vertex-disjoint temporal paths is equal to 1.