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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).

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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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Authors and Affiliations

  1. Department of Computer Science, University of Regina, Regina, SK, S4S 0A2, Canada

    Boting Yang

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  1. Boting Yang
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Correspondence to Boting Yang.

Additional information

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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