Journal of Combinatorial Optimization

, Volume 37, Issue 3, pp 970–988 | Cite as

Minimum rank and zero forcing number for butterfly networks

  • Daniela Ferrero
  • Cyriac Grigorious
  • Thomas KalinowskiEmail author
  • Joe Ryan
  • Sudeep Stephen


Zero forcing is a graph propagation process introduced in quantum physics and theoretical computer science, and closely related to the minimum rank problem. The minimum rank of a graph is the smallest possible rank over all matrices described by a given network. We use this relationship to determine the minimum rank and the zero forcing number of butterfly networks, concluding they present optimal properties in regards to both problems.


Zero forcing Minimum rank of graphs Butterfly network 

Mathematics Subject Classification

05C96 05C57 94C15 



We would like to thank an anonymous reviewer for carefully reading a previous version of the paper and providing a large number of insightful comments which were incredibly helpful in clarifying the presentation of our arguments.


  1. AIM Minimum Rank—Special Graphs Work Group, Barioli F, Barrett W, Butle S, Cioabă SM, Cvetković D, Fallat SM, Godsil C, Haemers W, Hogben L, Mikkelson R, Narayan S, Pryporova O, Sciriha I, So W, Stevanović D, van der Holst H, Van der Meulen K, Wangsness A (2008) Zero forcing sets and the minimum rank of graphs. Linear Algebra Appl 428(7):1628–1648Google Scholar
  2. Baldwin TL, Mili L, Boisen MB, Adapa R (1993) Power system observability with minimal phasor measurement placement. IEEE Trans Power Syst 8(2):707–715CrossRefGoogle Scholar
  3. Benson KF, Ferrero D, Flagg M, Furst V, Hogben L, Vasilevska V, Wissman B (2017) Zero forcing and power domination for graph products. Australas J Comb 70(2):221–235MathSciNetzbMATHGoogle Scholar
  4. Bienstock D (1991) Graph searching, path-width, tree-width and related problems (a survey). In: Roberts FS, Hwang FK, Monma CL (eds) Reliability of computer and communication network. DIMACS series in discrete mathematics and theoretical computer science, vol 5. American Mathematical Society, Providence, pp 33–49Google Scholar
  5. Bienstock D, Seymour P (1991) Monotonicity in graph searching. J Algorithms 12(2):239–245MathSciNetCrossRefzbMATHGoogle Scholar
  6. Burgarth D, Giovannetti V (2007) Full control by locally induced relaxation. Phys Rev Lett 99(10):100501CrossRefGoogle Scholar
  7. Dendris ND, Kirousis LM, Thilikos DM (1997) Fugitive-search games on graphs and related parameters. Theor Comput Sci 172(1–2):233–254MathSciNetCrossRefzbMATHGoogle Scholar
  8. Dobrev S, Flocchini P, Královič R, Ružička P, Prencipe G, Santoro N (2006) Black hole search in common interconnection networks. Networks 47(2):61–71MathSciNetCrossRefzbMATHGoogle Scholar
  9. Dyer D, Yang B, Yaşar Ö (2008) On the fast searching problem, algorithmic aspects in information and management. Lecture notes in computer science. Springer, Berlin, pp 143–154CrossRefzbMATHGoogle Scholar
  10. Fallat SM, Hogben L (2007) The minimum rank of symmetric matrices described by a graph: a survey. Linear Algebra Appl 426(2–3):558–582MathSciNetCrossRefzbMATHGoogle Scholar
  11. Fazel M, Hindi H, Boyd S (2004) Rank minimization and applications in system theory. In: Proceedings of the 2004 American control conference, vol 4. IEEE, pp 3273–3278Google Scholar
  12. Ferrero D, Hogben L, Kenter FHJ, Young M (2016) Note on power propagation time and lower bounds for the power domination number. J Comb Optim 34(3):736–741MathSciNetCrossRefzbMATHGoogle Scholar
  13. Haynes TW, Hedetniemi SM, Hedetniemi ST, Henning MA (2002) Domination in graphs applied to electric power networks. SIAM J Discrete Math 15(4):519–529MathSciNetCrossRefzbMATHGoogle Scholar
  14. Hogben L, Huynh M, Kingsley N, Meyer S, Walker S, Young M (2012) Propagation time for zero forcing on a graph. Discrete Appl Math 160(13–14):1994–2005MathSciNetCrossRefzbMATHGoogle Scholar
  15. Hogben L, Barrett W, Grout J, van der Holst H, Rasmussen K, Smith A (2016) AIM minimum rank graph catalog.
  16. Huang L-H, Chang GJ, Yeh H-G (2010) On minimum rank and zero forcing sets of a graph. Linear Algebra Appl 432(11):2961–2973MathSciNetCrossRefzbMATHGoogle Scholar
  17. Kirousis Lefteris M, Kranakis E, Krizanc D, Stamatiou YC (2000) Locating information with uncertainty in fully interconnected networks. In: International symposium on distributed computing, lecture notes in computer science, vol 1914. Springer, Berlin, pp 283–296Google Scholar
  18. Severini S (2008) Nondiscriminatory propagation on trees. J Phys A Math Theor 41(48):482002MathSciNetCrossRefzbMATHGoogle Scholar
  19. Yang B (2013) Fast–mixed searching and related problems on graphs. Theor Comput Sci 507:100–113MathSciNetCrossRefzbMATHGoogle Scholar

Copyright information

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

Authors and Affiliations

  1. 1.Department of MathematicsTexas State UniversitySan MarcosUSA
  2. 2.School of Mathematical and Physical SciencesUniversity of NewcastleCallaghanAustralia
  3. 3.Graduate SchoolKings College LondonLondonUK
  4. 4.School of Science and TechnologyUniversity of New EnglandArmidaleAustralia
  5. 5.School of Mathematical SciencesNational Institute of Science Education and ResearchBhubaneswarIndia

Personalised recommendations