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Temporal graphs are graphs whose topology is subject to discrete changes over time. Given a static underlying graph $G$, a temporal graph is represented by assigning a set of integer time-labels to every edge $e$ of $G$, indicating the…

Discrete Mathematics · Computer Science 2020-09-30 George B. Mertzios , Hendrik Molter , Rolf Niedermeier , Viktor Zamaraev , Philipp Zschoche

In this paper we initiate the study of the temporal graph realization problem with respect to the fastest path durations among its vertices, while we focus on periodic temporal graphs. Given an $n \times n$ matrix $D$ and a $\Delta \in…

Data Structures and Algorithms · Computer Science 2024-05-02 Nina Klobas , George B. Mertzios , Hendrik Molter , Paul G. Spirakis

A temporal graph is a graph in which edges are assigned a time label. Two nodes u and v of a temporal graph are connected one to the other if there exists a path from u to v with increasing edge time labels. We consider the problem of…

Data Structures and Algorithms · Computer Science 2023-04-04 Alkida Balliu , Filippo Brunelli , Pierluigi Crescenzi , Dennis Olivetti , Laurent Viennot

Reachability questions are one of the most fundamental algorithmic primitives in temporal graphs -- graphs whose edge set changes over discrete time steps. A core problem here is the NP-hard Short Restless Temporal Path: given a temporal…

Data Structures and Algorithms · Computer Science 2022-03-31 Philipp Zschoche

A temporal graph can be represented by a graph with an edge labelling, such that an edge is present in the network if and only if the edge is assigned the corresponding time label. A journey is a labelled path in a temporal graph such that…

Computational Complexity · Computer Science 2023-12-13 Esteban Christiann , Eric Sanlaville , Jason Schoeters

Let $G=(V,E)$ be an undirected graph on $n$ vertices and $\lambda:E\to 2^{\mathbb{N}}$ a mapping that assigns to every edge a non-empty set of integer labels (times). Such a graph is {\em temporally connected} if a path exists with…

Discrete Mathematics · Computer Science 2021-04-29 Arnaud Casteigts , Joseph G. Peters , Jason Schoeters

Modern, inherently dynamic systems are usually characterized by a network structure, i.e. an underlying graph topology, which is subject to discrete changes over time. Given a static underlying graph $G$, a temporal graph can be represented…

Computational Complexity · Computer Science 2019-08-13 Eleni C. Akrida , George B. Mertzios , Paul G. Spirakis , Viktor Zamaraev

In dynamic graphs, edges may be added or deleted in each synchronous round. Various connectivity models exist based on constraints on these changes. One well-known model is the $T$-Interval Connectivity model, where the graph remains…

Distributed, Parallel, and Cluster Computing · Computer Science 2025-04-14 Ashish Saxena , Kaushik Mondal

Temporal graphs represent interactions between entities over the time. These interactions may be direct (a contact between two nodes at some time instant), or indirect, through sequences of contacts called temporal paths (journeys).…

Data Structures and Algorithms · Computer Science 2021-04-01 Luiz F. Afra Brito , Marcelo Albertini , Arnaud Casteigts , Bruno A. N. Travençolo

A temporal graph is a graph whose edges only appear at certain points in time. Reachability in these graphs is defined in terms of paths that traverse the edges in chronological order (temporal paths). This form of reachability is neither…

Computational Complexity · Computer Science 2024-07-01 Arnaud Casteigts , Nils Morawietz , Petra Wolf

A temporal graph is a graph whose edges appear at certain points in time. These graphs are temporally connected (in class TC) if all vertices can reach each other by temporal paths (traversing the edges in chronological order). Reachability…

Discrete Mathematics · Computer Science 2026-04-21 Arnaud Casteigts , Timothée Corsini , Nils Morawietz

While a natural fit for modeling and understanding mobile networks, time-varying graphs remain poorly understood. Indeed, many of the usual concepts of static graphs have no obvious counterpart in time-varying ones. In this paper, we…

Networking and Internet Architecture · Computer Science 2012-08-01 John Whitbeck , Marcelo Dias de Amorim , Vania Conan , Jean-Loup Guillaume

Connectivity of temporal graphs has been widely studied both as graph theory and as gossip theory. In particular, it is well known that in order to connect every vertex to every other, a temporal graph needs to have at least $2n-4$ edges…

Data Structures and Algorithms · Computer Science 2026-05-01 Thomas Bellitto , Jules Bouton Popper , Justine Cauvi , Bruno Escoffier , Raphaëlle Maistre-Matus

A temporal graph is a graph in which vertices communicate with each other at specific time, e.g., $A$ calls $B$ at 11 a.m. and talks for 7 minutes, which is modeled by an edge from $A$ to $B$ with starting time "11 a.m." and duration "7…

Databases · Computer Science 2016-01-26 Huanhuan Wu , Yuzhen Huang , James Cheng , Jinfeng Li , Yiping Ke

The Temporal Graph Exploration problem (TEXP) takes as input a temporal graph, i.e., a sequence of graphs $(G_i)_{i\in \mathbb{N}}$ on the same vertex set, and asks for a walk of shortest length visiting all vertices, where the $i$-th step…

Discrete Mathematics · Computer Science 2025-08-06 Samuel Baguley , Andreas Göbel , Nicolas Klodt , George Skretas , John Sylvester , Viktor Zamaraev

Temporal graphs arise when modeling interactions that evolve over time. They usually come in several flavors, depending on the number of parameters used to describe the temporal aspects of the interactions: time of appearance, duration,…

Data Structures and Algorithms · Computer Science 2026-01-26 Guillaume Aubian , Filippo Brunelli , Feodor F Dragan , Guillaume Ducoffe , Michel Habib , Allen Ibiapina , Laurent Viennot

The fully dynamic transitive closure problem asks to maintain reachability information in a directed graph between arbitrary pairs of vertices, while the graph undergoes a sequence of edge insertions and deletions. The problem has been…

Data Structures and Algorithms · Computer Science 2020-02-04 Kathrin Hanauer , Monika Henzinger , Christian Schulz

\emph{Temporal graphs} are a generalisation of (static) graphs, defined by a sequence of \emph{snapshots}, each a static graph defined over a common set of vertices. \emph{Exploration} problems are one of the most fundamental and most…

Data Structures and Algorithms · Computer Science 2026-02-24 Duncan Adamson , Paul G Spirakis

In this work we consider temporal graphs, i.e. graphs, each edge of which is assigned a set of discrete time-labels drawn from a set of integers. The labels of an edge indicate the discrete moments in time at which the edge is available. We…

Data Structures and Algorithms · Computer Science 2013-10-30 Paul G. Spirakis , Eleni Ch. Akrida

We consider the problem of assigning appearing times to the edges of a digraph in order to maximize the (average) temporal reachability between pairs of nodes. Motivated by the application to public transit networks, where edges cannot be…

Discrete Mathematics · Computer Science 2025-01-22 Filippo Brunelli , Pierluigi Crescenzi , Laurent Viennot