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Distance-edge-monitoring sets of networks

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Abstract

It is important to be able to monitor the network and detect this failure when a connection (an edge) fails. For a vertex set M and an edge e of the graph G, let P(M, e) be the set of pairs (x, y) with a vertex x of M and a vertex y of V(G) such that e belongs to all shortest paths between x and y. A vertex set M of the graph G is distance-edge-monitoring set if every edge e of G is monitored by some vertex of M, that is, the set P(M, e) is nonempty. The distance-edge-monitoring number of a graph G, recently introduced by Foucaud, Kao, Klasing, Miller, and Ryan, is defined as the smallest size of distance-edge-monitoring sets of G. In this paper, we determine the bounds of the distance-edge-monitoring number of grid-based pyramids and the exact value of distance-edge-monitoring number for M(t)-graph and Sierpiński-type graphs. We also compare the distance-edge-monitoring set with average degree, the size of edge set and the size of vertex set of G, where G is M(t)-graph or Sierpiński-type graphs.

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Funding

Supported by the National Science Foundation of China (Nos. 11601254, 11551001) and the Qinghai Key Laboratory of Internet of Things Project (2017-ZJ-Y21).

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Authors and Affiliations

  1. College of Science, University of Shanghai for Science and Technology, Shanghai, 200093, China

    Gang Yang & Changxiang He

  2. School of Mathematics and Statistics, Qinghai Normal University, Xining, Qinghai, 810008, China

    Jiannan Zhou

  3. JSPS International Research Fellow, Faculty of Environment and Information Sciences, Yokohama National University, 79-2 Tokiwadai, Hodogaya-ku, Yokohama, 240-8501, Japan

    Yaping Mao

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  1. Gang Yang
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Correspondence to Yaping Mao.

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Yang, G., Zhou, J., He, C. et al. Distance-edge-monitoring sets of networks. Acta Informatica (2024). https://doi.org/10.1007/s00236-024-00453-z

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