, Volume 80, Issue 6, pp 1857–1889 | Cite as

Deleting Edges to Restrict the Size of an Epidemic: A New Application for Treewidth

  • Jessica Enright
  • Kitty Meeks


Motivated by applications in network epidemiology, we consider the problem of determining whether it is possible to delete at most k edges from a given input graph (of small treewidth) so that the resulting graph avoids a set \(\mathcal {F}\) of forbidden subgraphs; of particular interest is the problem of determining whether it is possible to delete at most k edges so that the resulting graph has no connected component of more than h vertices, as this bounds the worst-case size of an epidemic. While even this special case of the problem is NP-complete in general (even when \(h=3\)), we provide evidence that many of the real-world networks of interest are likely to have small treewidth, and we describe an algorithm which solves the general problem in time \(2^{O(|\mathcal {F}|w^r)}n\) on an input graph having n vertices and whose treewidth is bounded by a fixed constant w, if each of the subgraphs we wish to avoid has at most r vertices. For the special case in which we wish only to ensure that no component has more than h vertices, we improve on this to give an algorithm running in time \(O((wh)^{2w}n)\), which we have implemented and tested on real datasets based on cattle movements.


Edge-deletion Treewidth Network epidemiology Graph contagion 



The authors would like to thank the following: Ivaylo Valkov for his assistance in developing an initial implementation of this algorithm as part of a summer research project, and the Engineering and Physical Sciences Research Council for providing funding for this summer project; EPIC: Scotland’s Centre of Expertise on Animal Disease Outbreaks, which supported JE for part of her work on this project; the Royal Society of Edinburgh which supported KM for part of her work on this project through a Personal Research Fellowship funded by the Scottish Government; Ciaran McCreesh and Patrick Prosser for their assistance in developing the CP formulation against which we compared the performance of our algorithm.


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Copyright information

© Springer Science+Business Media New York 2017

Authors and Affiliations

  1. 1.Computing Science and MathematicsUniversity of StirlingStirlingUK
  2. 2.School of Computing Science, Sir Alwyn Williams BuildingUniversity of GlasgowGlasgowUK

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