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Lower bounds for positive semidefinite zero forcing and their applications

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Abstract

The positive semidefinite zero forcing number of a graph is a parameter that is important in the study of minimum rank problems. In this paper, we focus on the algorithmic aspects of computing this parameter. We prove that it is NP-complete to find the positive semidefinite zero forcing number of a given graph, and this problem remains NP-complete even for graphs with maximum vertex degree 7. We present a linear time algorithm for computing the positive semidefinite zero forcing number of generalized series–parallel graphs. We introduce the constrained tree cover number and apply it to improve lower bounds for positive semidefinite zero forcing. We also give formulas for the constrained tree cover number and the tree cover number on graphs with special structures.

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Notes

  1. There should not be an edge between \(a'\) and \(b'\) on the third graph in Fig. 4 of Yang (2013).

References

  • AIM Minimum Rank-Special Graphs Work Group (2008) Zero forcing sets and the minimum rank of graphs. Linear Algebra Appl 428(7):1628–1648

    Article  MathSciNet  Google Scholar 

  • Barioli F, Barrett W, Fallat S, Hall HT, Hogben L, Shader B, van den Driessche P, van der Holst H (2013) Parameters related to tree-width, zero forcing, and maximum nullity of a graph. J Graph Theory 72:146–177

    Article  MathSciNet  MATH  Google Scholar 

  • Barioli F, Barrett W, Fallat S, Hall HT, Hogben L, Shader B, van den Driessche P, van der Holst H (2010) Zero forcing parameters and minimum rank problems. Linear Algebra Appl 433(2):401–411

    Article  MathSciNet  MATH  Google Scholar 

  • Barioli F, Fallat S, Mitchell L, Narayan S (2011) Minimum semidefinite rank of outerplanar graphs and the tree cover number. Electron J Linear Algebra 22:10–21

    Article  MathSciNet  MATH  Google Scholar 

  • Bienstock D, Seymour P (1991) Monotonicity in graph searching. J Algorithms 12:239–245

    Article  MathSciNet  MATH  Google Scholar 

  • Booth M, Hackney P, Harris B, Johnson CR, Lay M, Mitchell LH, Narayan SK, Pascoe A, Steinmetz K, Sutton BD, Wang W (2008) On the minimum rank among positive semidefinite matrices with a given graph. SIAM J Matrix Anal Appl 30:731–740

    Article  MathSciNet  MATH  Google Scholar 

  • Burgarth D, Giovannetti V (2007) Full control by locally induced relaxation. Phys Rev Lett 99(10):100–501

    Article  Google Scholar 

  • Duffin RJ (1965) Topology of series parallel networks. J Math Appl 10:303–318

    MathSciNet  MATH  Google Scholar 

  • Dyer D, Yang B, Yaşar O (2008) On the fast searching problem. In: Proceedings of the 4th International Conference on Algorithmic Aspects in Information and Management (AAIM’08), Lecture notes in Computer Science, vol 5034. Springer, New York, pp 143–154

  • Ekstrand J, Erickson C, Hall HT, Hay D, Hogben L, Johnson R, Kingsley N, Osborne S, Peters T, Roat J, Ross A, Row D, Warnberg N, Young M (2013) Positive semidefinite zero forcing. Linear Algebra Appl 439:1862–1874

    Article  MathSciNet  MATH  Google Scholar 

  • Ekstrand J, Erickson C, Hay D, Hogben L, Roat J (2012) Note on positive semidefinite maximum nullity and positive semidefinite zero forcing number of partial \(2\)-trees. Electron J Linear Algebra 23:79–87

    Article  MathSciNet  MATH  Google Scholar 

  • Fallat S, Meagher K, Yang B (2015) On the complexity of the positive semidefinite zero forcing number. Linear Algebra Appl doi:10.1016/j.laa.2015.03.011

  • Hopcroft J, Tarjan R (1973) Efficient algorithms for graph manipulation. Commun ACM 16(6):372–378

    Article  Google Scholar 

  • Kirousis L, Papadimitriou C (1986) Searching and pebbling. Theor Comput Sci 47:205–218

    Article  MathSciNet  MATH  Google Scholar 

  • Kratochvíl J, Tuza Z (2002) On the complexity of bicoloring clique hypergraphs of graphs. J Algorithms 45:40–54

    Article  MathSciNet  MATH  Google Scholar 

  • Megiddo N, Hakimi S, Garey M, Johnson D, Papadimitriou C (1988) The complexity of searching a graph. J ACM 35:18–44

    Article  MathSciNet  MATH  Google Scholar 

  • Robertson N, Seymour P (1983) Graph minors I: excluding a forest. J Comb Theor B 35:39–61

    Article  MathSciNet  MATH  Google Scholar 

  • Robertson N, Seymour P (1984) Graph minors III: planar tree-width. J Comb Theor B 36:49–64

    Article  MathSciNet  MATH  Google Scholar 

  • Valdes J, Tarjan RE, Lawler EL (1982) The recognition of series parallel digraphs, SIAM J Comput 11: 289–313, In: Proc. 11th ACM Symp. Theory of Computing pp 1–12, 1979

  • West DB (2001) Introduction to graph theory, 2nd edn. Prentice Hall, Upper Saddle River

    Google Scholar 

  • Yang B (2013) Fast-mixed searching and related problems on graphs. Theor Comput Sci 507(7):100–113

    Article  MathSciNet  MATH  Google Scholar 

  • Yang B (2007) Strong-mixed searching and pathwidth. J Comb Optim 13:47–59

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgments

The author would like to thank the anonymous referees for their valuable comments and suggestions, which improved the presentation of this paper. The author would also like to thank Shaun Fallat and Karen Meagher for discussions on the complexity of positive semidefinite zero forcing.

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Correspondence to Boting Yang.

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Research supported in part by an NSERC Discovery Research Grant, Application No. RGPIN-2013-261290.

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Yang, B. Lower bounds for positive semidefinite zero forcing and their applications. J Comb Optim 33, 81–105 (2017). https://doi.org/10.1007/s10878-015-9936-0

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  • DOI: https://doi.org/10.1007/s10878-015-9936-0

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