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Strong Geodetic Number of Graphs and Connectivity

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Abstract

A recent variation of the classical geodetic problem, the strong geodetic problem, is defined as follows. If G is a graph, then \(\mathrm{sg}(G)\) is the cardinality of a smallest vertex subset S, such that one can assign a fixed geodesic to each pair \(\{x,y\}\subseteq S\) so that these \({|S|\atopwithdelims ()2}\) geodesics cover all the vertices of G. In this paper, we first give some bounds for strong geodetic number in terms of diameter, connectivity, respectively. Next, we show that \(2\le \mathrm{sg}(G)\le n\) for a connected graph G of order n, and graphs with \(\mathrm{sg}(G)=2,n-1,n\) are characterized, respectively. In the end, we investigate the Nordhaus–Gaddum-type problem and extremal problems for strong geodetic number.

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Acknowledgements

The authors are very grateful to the referees for their valuable comments and suggestions, which improved the presentation of this paper.

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Authors and Affiliations

  1. College of Science, China Jiliang University, Hangzhou, 310018, China

    Zhao Wang

  2. Department of Mathematics, Qinghai Normal University, Xining, 810008, Qinghai, China

    Yaping Mao

  3. Department of Mathematics, Clayton State University, Morrow, GA, 30260, USA

    Colton Magnant

  4. Academy of Plateau Science and Sustainability, Xining, 810008, Qinghai, China

    Yaping Mao & Colton Magnant

  5. School of Computer, Qinghai Normal University, Xining, 810008, Qinghai, China

    Huifen Ge

Authors
  1. Zhao Wang
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  2. Yaping Mao
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  3. Huifen Ge
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  4. Colton Magnant
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Corresponding author

Correspondence to Yaping Mao.

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Xueliang Li.

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Supported by the National Science Foundation of China (Nos. 11601254, 11551001, 11161037, 11461054) and the Science Found of Qinghai Province (No. 2019-ZJ-921) and the Qinghai Key Laboratory of Internet of Things Project (2017-ZJ-Y21).

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Wang, Z., Mao, Y., Ge, H. et al. Strong Geodetic Number of Graphs and Connectivity. Bull. Malays. Math. Sci. Soc. 43, 2443–2453 (2020). https://doi.org/10.1007/s40840-019-00809-6

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  • DOI: https://doi.org/10.1007/s40840-019-00809-6

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