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On the Zero Forcing Number of Bijection Graphs

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Book cover Combinatorial Algorithms (IWOCA 2015)

Part of the book series: Lecture Notes in Computer Science ((LNTCS,volume 9538))

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Abstract

The zero forcing number of a graph is a graph parameter based on a color change process, which starts with a state, where all vertices are colored either black or white. In the next step a white vertex turns black, if it is the only white neighbor of some black vertex, and this step is then iterated. The zero forcing number Z(G) is defined as the minimum cardinality of a set S of black vertices such that the whole vertex set turns black.

In this paper we study Z(G) for the class of bijection graphs, where a bijection graph is a graph on 2n vertices that can be partitioned into two parts with n vertices each, joined by a perfect matching. For this class of graphs we show an upper bound for the zero forcing number and classify the graphs that attain this bound. We improve the general lower bound for the zero forcing number, which is \(Z(G)\ge \delta (G)\), for certain bijection graphs and use this improved bound to find the exact value of the zero forcing number for these graphs. This extends and strengthens results of Yi (2012) about the more restricted class of so called permutation graphs.

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Notes

  1. 1.

    Note that the term permutation graph is also used in a different way in literature, see [12].

References

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Authors and Affiliations

  1. Department of Mathematics and Mathematical Statistics, University of Umeå, 901-87, Umeå, Sweden

    Denys Shcherbak, Gerold Jäger & Lars-Daniel Öhman

Authors
  1. Denys Shcherbak
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  2. Gerold Jäger
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  3. Lars-Daniel Öhman
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Corresponding author

Correspondence to Gerold Jäger .

Editor information

Editors and Affiliations

  1. Dept. of Computer Science, University of Verona, Verona, Italy

    Zsuzsanna Lipták

  2. Dept. of Computing and Software, McMaster University, Hamilton, Canada

    William F. Smyth

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Shcherbak, D., Jäger, G., Öhman, LD. (2016). On the Zero Forcing Number of Bijection Graphs. In: Lipták, Z., Smyth, W. (eds) Combinatorial Algorithms. IWOCA 2015. Lecture Notes in Computer Science(), vol 9538. Springer, Cham. https://doi.org/10.1007/978-3-319-29516-9_28

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  • DOI: https://doi.org/10.1007/978-3-319-29516-9_28

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  • Publisher Name: Springer, Cham

  • Print ISBN: 978-3-319-29515-2

  • Online ISBN: 978-3-319-29516-9

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