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Temporal Reachability Dominating Sets: Contagion in Temporal Graphs

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Algorithmics of Wireless Networks (ALGOWIN 2023)

Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 14061))

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Abstract

SARS-CoV-2 was independently introduced to the UK at least 1300 times by June 2020. Given a population with dynamic pairwise connections, we ask if the entire population could be (indirectly) infected by a small group of k initially infected individuals. We formalise this problem as the Temporal Reachability Dominating Set (TaRDiS) problem on temporal graphs. We provide positive and negative parameterized complexity results in four different parameters: the number k of initially infected, the lifetime \(\tau \) of the graph, the number of locally earliest edges in the graph, and the treewidth of the footprint graph \(\mathcal {G}_\downarrow \). We additionally introduce and study the MaxMinTaRDiS problem, which can be naturally expressed as scheduling connections between individuals so that a population needs to be infected by at least k individuals to become fully infected. Interestingly, we find a restriction of this problem to correspond exactly to the well-studied Distance-3 Independent Set problem on static graphs.

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Notes

  1. 1.

    For a given \(n\in \mathbb N^{>0}\) we denote by [n] the set \(\{1,2,\ldots , n\}\).

  2. 2.

    In Casteigts et al.’s work [12], a temporal graph is happy if it is both simple (only one time per edge) and proper (every vertex incident to at most 1 edge at a time); under our definition, all temporal graphs are simple.

  3. 3.

    When \(\mathcal {G}\) is clear from context, we write \(R_u\) for \(R_u(\mathcal {G})\) and SR(T, v) for \(SR(T,\mathcal {G},v)\).

  4. 4.

    If \(e=(u,v)\) is an edge, u is said to be incident to vertex v and to edge e. Also, u is incident to a set S of vertices or edges if and only if it is incident to some element of S.

  5. 5.

    In this work, we say a temporal graph is planar if and only if its footprint is planar.

  6. 6.

    We say that two problems X and Y are equivalent if they have the same language - that is, an instance I is a yes-instance of X if and only if the same instance I is a yes-instance of Y. Where X has a language consisting of triples \((G,k,\tau )\) and Y has a language of tuples (G, k), we may say that Y is equivalent to X with \(\tau \) fixed to some value.

  7. 7.

    Intuitively, the treewidth \(\text {tw}(G)\) of a graph G represents how “treelike” G is. We refer the interested reader to Chap. 7 in [17].

  8. 8.

    In this work, by “each variant” we refer to the Strict, Nonstrict and Happy variants of the problems introduced in Sect. 1.1.

  9. 9.

    This mirrors Lemma 54 from [5].

  10. 10.

    The result in Theorem 1 generalizes to Strict TaRDiS and Nonstrict TaRDiS; D3IS and Dominating Set are NP-complete on planar graphs.

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Authors and Affiliations

  1. Durham University, Durham, England, UK

    David C. Kutner

  2. University of Glasgow, Glasgow, Scotland, UK

    Laura Larios-Jones

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  1. Toronto Metropolitan University, Toronto, ON, Canada

    Konstantinos Georgiou

  2. Carleton University, Ottawa, ON, Canada

    Evangelos Kranakis

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Kutner, D.C., Larios-Jones, L. (2023). Temporal Reachability Dominating Sets: Contagion in Temporal Graphs. In: Georgiou, K., Kranakis, E. (eds) Algorithmics of Wireless Networks. ALGOWIN 2023. Lecture Notes in Computer Science, vol 14061. Springer, Cham. https://doi.org/10.1007/978-3-031-48882-5_8

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