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Realization of Temporally Connected Graphs Based on Degree Sequences

Authors Arnaud Casteigts , Michelle Döring , Nils Morawietz

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ACM Subject Classification
  • Theory of computation → Graph algorithms analysis
  • Mathematics of computing → Discrete mathematics
  • Networks → Network design and planning algorithms
Keywords
  • temporal paths
  • gossiping
  • (multi)graphical degree sequences
  • edge-disjoint spanning trees
  • linear time algorithms

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Abstract

Given an undirected graph G, the problem of deciding whether G admits a simple and proper time-labeling that makes it temporally connected is known to be NP-hard (Göbel et al., 1991). In this article, we relax this problem and ask whether a given degree sequence can be realized as a temporally connected graph. Our main results are a complete characterization of the feasible cases, and a recognition algorithm that runs in 𝒪(n) time for graphical degree sequences (realized as simple temporal graphs) and in 𝒪(n+m) time for multigraphical degree sequences (realized as non-simple temporal graphs, where the number of time labels on an edge corresponds to the multiplicity of the edge in the multigraph). In fact, these algorithms can be made constructive at essentially no cost. Namely, we give a constructive 𝒪(n+m) time algorithm that outputs, for a given (multi)graphical degree sequence 𝐝, a temporally connected graph whose underlying (multi)graph is a realization of 𝐝, if one exists.

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Arnaud Casteigts, Michelle Döring, and Nils Morawietz. Realization of Temporally Connected Graphs Based on Degree Sequences. In 36th International Symposium on Algorithms and Computation (ISAAC 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 359, pp. 17:1-17:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.ISAAC.2025.17

Author Details

Arnaud Casteigts
  • Department of Computer Science, University of Geneva, Switzerland
  • LaBRI, Université de Bordeaux, France
Michelle Döring
  • Hasso Plattner Institute, University of Potsdam, Germany
Nils Morawietz
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
  • LaBRI, Université de Bordeaux, France

Funding

  • Casteigts, Arnaud: Supported by the Swiss NSF, project RECAPT (200021-236640).
  • Döring, Michelle: German Federal Ministry for Education and Research (BMBF) through the project "KI Servicezentrum Berlin Brandenburg" (01IS22092).
  • Morawietz, Nils: Supported by the French ANR, project ANR-22-CE48-0001 (TEMPOGRAL).

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