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International Workshop on Algorithms and Models for the Web-Graph

WAW 2023: Algorithms and Models for the Web Graph pp 68–82Cite as

Establishing Herd Immunity is Hard Even in Simple Geometric Networks

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Part of the book series: Lecture Notes in Computer Science ((LNCS,volume 13894))

Abstract

We study the following model of disease spread in a social network. In the beginning, all individuals are either infected or healthy. Next, in discrete rounds, the disease spreads in the network from infected to healthy individuals such that a healthy individual gets infected if and only if a sufficient number of its direct neighbours are already infected.

We represent the social network as a graph. Inspired by the real-world restrictions in the current epidemic, especially by social and physical distancing requirements, we restrict ourselves to networks that can be represented as geometric intersection graphs.

We show that finding a minimal vertex set of initially infected individuals to spread the disease in the whole network is computationally hard, already on unit disk graphs. Hence, to provide some algorithmic results, we focus ourselves on simpler geometric graph families, such as interval graphs and grid graphs.

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Notes

  1. 1.

    Rectilinear embedding of a planar graph with maximum degree 4 is planar embedding with vertices at integer coordinates and edges are drawn so that they are made up of line segments of the form \(x=i\) or \(y=j\), where \(i,j\in \mathbb {Z}\) [34].

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Acknowledgments

The authors acknowledge the support of the Czech Science Foundation Grant No. 22-19557S. MD and ŠS were additionally supported by the Grant Agency of the Czech Technical University in Prague, grant No. SGS23/205/OHK3/3T/18. MD was also supported by the Student Summer Research Program 2021 of FIT CTU in Prague.

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Authors and Affiliations

  1. Department of Theoretical Computer Science, Faculty of Information Technology, Czech Technical University in Prague, Prague, Czech Republic

    Michal Dvořák, Dušan Knop & Šimon Schierreich

Authors
  1. Michal Dvořák
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  2. Dušan Knop
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  3. Šimon Schierreich
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Corresponding author

Correspondence to Šimon Schierreich .

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Editors and Affiliations

  1. Tutte Institute for Mathematics and Computing, Ottawa, ON, Canada

    Megan Dewar

  2. Toronto Metropolitan University, Toronto, ON, Canada

    Paweł Prałat

  3. SGH Warsaw School of Economics, Warsaw, Poland

    Przemysław Szufel

  4. Tutte Institute for Mathematics and Computing, Ottawa, ON, Canada

    François Théberge

  5. SGH Warsaw School of Economics, Warsaw, Poland

    Małgorzata Wrzosek

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Dvořák, M., Knop, D., Schierreich, Š. (2023). Establishing Herd Immunity is Hard Even in Simple Geometric Networks. In: Dewar, M., Prałat, P., Szufel, P., Théberge, F., Wrzosek, M. (eds) Algorithms and Models for the Web Graph. WAW 2023. Lecture Notes in Computer Science, vol 13894. Springer, Cham. https://doi.org/10.1007/978-3-031-32296-9_5

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  • DOI: https://doi.org/10.1007/978-3-031-32296-9_5

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