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Journal of Combinatorial Optimization

, Volume 37, Issue 3, pp 935–956 | Cite as

Restricted power domination and zero forcing problems

  • Chassidy Bozeman
  • Boris BrimkovEmail author
  • Craig Erickson
  • Daniela Ferrero
  • Mary Flagg
  • Leslie Hogben
Article

Abstract

Power domination in graphs arises from the problem of monitoring an electric power system by placing as few measurement devices in the system as possible. A power dominating set of a graph is a set of vertices that observes every vertex in the graph, following a set of rules for power system monitoring. A practical problem of interest is to determine the minimum number of additional measurement devices needed to monitor a power network when the network is expanded and the existing devices remain in place. In this paper, we study the problem of finding the smallest power dominating set that contains a given set of vertices X. We also study the related problem of finding the smallest zero forcing set that contains a given set of vertices X. The sizes of such sets in a graph G are respectively called the restricted power domination number and restricted zero forcing number of G subject to X. We derive several tight bounds on the restricted power domination and zero forcing numbers of graphs, and relate them to other graph parameters. We also present exact and algorithmic results for computing the restricted power domination number, including integer programs for general graphs and a linear time algorithm for graphs with bounded treewidth. We also use restricted power domination to obtain a parallel algorithm for finding minimum power dominating sets in trees.

Keywords

Power domination Restricted power domination Zero forcing Restricted zero forcing 

Mathematics Subject Classification

05C69 05C50 05C57 94C15 

Notes

Acknowledgements

This research began at the American Institute of Mathematics workshop Zero forcing and its applications with support from National Science Foundation, DMS-1128242. The work of BB is supported in part by the National Science Foundation, Grant No. 1450681. We thank AIM and NSF for their support. We also thank the referees for their careful reading and helpful comments, which have improved the paper.

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Copyright information

© Springer Science+Business Media, LLC, part of Springer Nature 2018

Authors and Affiliations

  • Chassidy Bozeman
    • 1
  • Boris Brimkov
    • 2
    Email author
  • Craig Erickson
    • 3
  • Daniela Ferrero
    • 4
  • Mary Flagg
    • 5
  • Leslie Hogben
    • 1
    • 6
  1. 1.Department of MathematicsIowa State UniversityAmesUSA
  2. 2.Department of Computational and Applied MathematicsRice UniversityHoustonUSA
  3. 3.Saint PaulUSA
  4. 4.Department of MathematicsTexas State UniversitySan MarcosUSA
  5. 5.Department of Mathematics, Computer Science and Cooperative EngineeringUniversity of St. ThomasHoustonUSA
  6. 6.American Institute of MathematicsSan JoseUSA

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