Snapshot disjointness in temporal graphs

TL;DR

This paper introduces and studies snapshot disjointness in temporal graphs, analyzing its computational complexity, characterizing Mengerian graphs, and exploring related separation problems.

Contribution

It defines snapshot disjointness, proves complexity results, characterizes Mengerian graphs for time, and provides algorithms for recognition and separation problems.

Findings

Snapshot disjointness is W[1]-hard and XP-time solvable.

Polynomial-time recognition of Mengerian graphs for time.

Deciding minimal multiedge separation is NP-complete.

Abstract

In the study of temporal graphs, only paths respecting the flow of time are relevant. In this context, many concepts of walks disjointness were proposed over the years, and the validity of Menger's Theorem, as well as the complexity of related problems, has been investigated. In this paper, we introduce and investigate a type of disjointness that is only time dependent. Two walks are said to be snapshot disjoint if they are not active in a same snapshot (also called timestep). The related paths and cut problems are then defined and proved to be W[1]-hard and XP-time solvable when parameterized by the size of the solution. Additionally, in the light of the definition of Mengerian graphs given by Kempe, Kleinberg and Kumar in their seminal paper (STOC'2000), we define a Mengerian graph for time as a graph $G$ that cannot form an example where Menger's Theorem does not hold in the context of snapshot disjointness. We then give a characterization in terms of forbidden structures and provide a polynomial-time recognition algorithm. Finally, we also prove that, given a temporal graph $(G,\lambda)$ and a pair of vertices $s,z\in V(G)$, deciding whether at most $h$ multiedges can separate $s$ from $z$ is NP-complete.