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Temporal Dominating Set and Temporal Vertex Cover Under the Lense of Degree Restrictions

Authors Anton Herrmann , Christian Komusiewicz , Nils Morawietz , Frank Sommer

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Subject Classification

ACM Subject Classification
  • Theory of computation → Parameterized complexity and exact algorithms
  • Theory of computation → Graph algorithms analysis
Keywords
  • NP-hard problem
  • FPT-algorithm
  • Treewidth
  • Color coding

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Abstract

We study the Temporal Dominating Set problem, in which one asks whether a temporal graph 𝒢 = (G₁,… , G_T) given as a sequence of snapshot graphs, over the same vertex set V, has a set S of temporal vertices of size at most k such that each vertex v of V is dominated by some w ∈ S in the snapshot that contains w. Additionally, we consider Temporal Partial Dominating Set, where one asks whether at least t (and not necessarily all) vertices of V can be dominated by S and a further generalization in which the solution may only contain a bounded number of temporal vertices from each snapshot.
We analyze how the complexity of Temporal (Partial) Dominating Set is influenced by the maximum snapshot degree and the structure of the underlying graph, the graph with vertex set V and whose edge set is the union of all snapshot edge sets. For example, we obtain a complexity dichotomy for the maximum snapshot degree and we show that Temporal Partial Dominating Set is fixed-parameter tractable for tw+Δ, where tw and Δ denote the treewidth and the maximum degree of the underlying graph of 𝒢, respectively. We also show which of our results transfer to the well-studied Temporal Vertex Cover problem. For example, we show that Temporal Vertex Cover is also fixed-parameter tractable for tw+Δ which substantially extends the previously known polynomial-time algorithms for the case that the underlying graph is a path or cycle.

Cite As Get BibTex

Anton Herrmann, Christian Komusiewicz, Nils Morawietz, and Frank Sommer. Temporal Dominating Set and Temporal Vertex Cover Under the Lense of Degree Restrictions. In 4th Symposium on Algorithmic Foundations of Dynamic Networks (SAND 2025). Leibniz International Proceedings in Informatics (LIPIcs), Volume 330, pp. 16:1-16:18, Schloss Dagstuhl – Leibniz-Zentrum für Informatik (2025) https://doi.org/10.4230/LIPIcs.SAND.2025.16

Author Details

Anton Herrmann
  • Algorithmics and Computational Complexity, Technische Universität Berlin, Germany
Christian Komusiewicz
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
Nils Morawietz
  • Institute of Computer Science, Friedrich Schiller University Jena, Germany
  • LaBRI, Université de Bordeaux, France
Frank Sommer
  • Institute of Logic and Computation, TU Wien, Austria

Funding

  • Morawietz, Nils: Partially supported by the French ANR, project ANR-22-CE48-0001 (TEMPOGRAL).
  • Sommer, Frank: Supported by the Alexander von Humboldt Foundation.

Acknowledgements

Some of the results of this work are also contained in the first author’s Master thesis [Anton Herrmann, 2024].

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