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.
For a given \(n\in \mathbb N^{>0}\) we denote by [n] the set \(\{1,2,\ldots , n\}\).
- 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.
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.
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.
In this work, we say a temporal graph is planar if and only if its footprint is planar.
- 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.
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.
In this work, by “each variant” we refer to the Strict, Nonstrict and Happy variants of the problems introduced in Sect. 1.1.
- 9.
This mirrors Lemma 54 from [5].
- 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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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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