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Designing Sparse Temporal Graphs Satisfying Connectivity Requirements

Authors Thomas Bellitto , Jules Bouton Popper , Justine Cauvi , Bruno Escoffier , Raphaëlle Maistre-Matus

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Subject Classification

ACM Subject Classification
  • Mathematics of computing → Graph algorithms
  • Mathematics of computing → Paths and connectivity problems
Keywords
  • Temporal Graphs
  • Connectivity
  • Gossiping
  • Network Design

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Abstract

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 where n is the number of vertices. This paper investigates the optimal number of edges required to satisfy partial connectivity requirements. We introduce the problem of Connectivity Request Satisfaction where we are given a directed graph that we call the request graph, where an arc from u to v means that we need to be able to go from u to v. Our goal is to build a temporal graph on the same vertex set with as few temporal edges as possible that would satisfy all the requests. When the graph we build is directed, we prove that the number of temporal arcs required is n-cc+dfvs where cc is the number of connected component of the request graph and dfvs is the size of its smallest directed feedback vertex set. It follows that the problem is NP-complete but inherits fixed parameter tractability properties of Directed Feedback Vertex Set. When the graph we build is undirected, we establish a characterization of strongly connected request graphs that admit a solution with n-1 edges: it is possible if and only if any set of pairwise non-vertex-disjoint closed walks all share a common vertex. We prove that this criteria can be tested in polynomial time.

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Thomas Bellitto, Jules Bouton Popper, Justine Cauvi, Bruno Escoffier, and Raphaëlle Maistre-Matus. Designing Sparse Temporal Graphs Satisfying Connectivity Requirements. In 5th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2026). Leibniz International Proceedings in Informatics (LIPIcs), Volume 373, pp. 15:1-15:17, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2026) https://doi.org/10.4230/LIPIcs.SAND.2026.15

Author Details

Thomas Bellitto
  • Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
Jules Bouton Popper
  • Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
Justine Cauvi
  • École Normale Supérieure de Lyon, Lyon, France
  • Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
Bruno Escoffier
  • Sorbonne Université, CNRS, LIP6, F-75005 Paris, France
Raphaëlle Maistre-Matus
  • Sorbonne Université, CNRS, LIP6, F-75005 Paris, France

Funding

  • Cauvi, Justine: This author has been partially supported by the French ANR project Tempogral ANR-22-CE48-0001

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