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The Threshold Dimension and Threshold Strong Dimension of a Graph: A Survey

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Research Trends in Graph Theory and Applications

Part of the book series: Association for Women in Mathematics Series ((AWMS,volume 25))

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Abstract

Let G be a connected graph and u, v and w vertices of G. Then w is said to resolve u and v if the distance from u to w does not equal the distance from v to w. If there is either a shortest u-w path that contains v or a shortest v-w path that contains u, then we say that w strongly resolves u and v. A set W of vertices of G is a resolving set (strong resolving set), if every pair of vertices of G is resolved (respectively, strongly resolved) by some vertex of W. A smallest resolving set (strong resolving set) of a graph is called a basis (respectively, a strong basis) and its cardinality, denoted β(G) (respectively, β s(G)), the metric dimension (respectively, the strong dimension) of G. The threshold dimension (respectively, threshold strong dimension) of a graph G, denoted τ(G) (respectively, τ s(G)), is the smallest metric dimension (respectively, strong dimension) among all graphs having G as a spanning subgraph. We survey results on the threshold dimension and threshold strong dimension and contrast these invariants. The paper concludes with several open problems for these parameters.

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Notes

  1. 1.

    The interval between two vertices x and y in a graph G is the collection of all vertices that lie on some shortest x–y path.

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Acknowledgements

The work of O. R. Oellermann was partially supported by an NSERC Grant CANADA, Grant number RGPIN-2016-05237.

Author information

Authors and Affiliations

  1. Department of Mathematics, New York City College of Technology, Brooklyn, NY, USA

    Nadia Benakli

  2. Department of Mathematical Sciences, University of Delaware, Newark, DE, USA

    Novi H. Bong

  3. Department of Mathematics and Statistics, The University of Winnipeg, Winnipeg, MB, Canada

    Shonda Dueck (Gosselin) & Ortrud R. Oellermann

  4. School of Mathematical and Statistical Sciences, Clemson University, Clemson, SC, USA

    Beth Novick

Authors
  1. Nadia Benakli
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  2. Novi H. Bong
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  3. Shonda Dueck (Gosselin)
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  4. Beth Novick
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  5. Ortrud R. Oellermann
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Corresponding author

Correspondence to Ortrud R. Oellermann .

Editor information

Editors and Affiliations

  1. Department of Mathematics, Texas State University, San Marcos, TX, USA

    Daniela Ferrero

  2. Department of Mathematics, Iowa State University, Ames, IA, USA

    Leslie Hogben

  3. Department of Mathematics, Brooklyn College and the Graduate Center, City University of New York, New York, NY, USA

    Sandra R. Kingan

  4. Department of Mathematics, Virginia Polytechnic Institute and State University, Blacksburg, VA, USA

    Gretchen L. Matthews

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Benakli, N., Bong, N.H., Dueck (Gosselin), S., Novick, B., Oellermann, O.R. (2021). The Threshold Dimension and Threshold Strong Dimension of a Graph: A Survey. In: Ferrero, D., Hogben, L., Kingan, S.R., Matthews, G.L. (eds) Research Trends in Graph Theory and Applications. Association for Women in Mathematics Series, vol 25. Springer, Cham. https://doi.org/10.1007/978-3-030-77983-2_4

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