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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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References

  1. Biló, D., Erlebach, T., Mihalák, M., Widmayer, P.: Discovery of network properties with all-shortest-paths queries. Theoret. Comput. Sci. 411(14–15), 1626–1637 (2010)

    Article  MathSciNet  Google Scholar 

  2. Bampas, E., Biló, D., Drovandi, G., Gualá, L., Klasing, R., Proietti, G.: Network verification via routing table queries. J. Comput. System Sci. 81(1), 234–248 (2015)

    Article  MathSciNet  Google Scholar 

  3. Beerliova, Z., Eberhard, F., Erlebach, T., Hall, A., Hoffmann, M., Mihalák, M., Ram, L.S.: Network discovery and verification. IEEE J. Sel. Areas Commun. 24(12), 2168–2181 (2006)

    Article  Google Scholar 

  4. Comellas, F., Miralles, A.: Modeling complex networks with self-similar outerplanar unclustered graphs. Phys. A 388(11), 2227–2233 (2009)

    Article  MathSciNet  Google Scholar 

  5. Comellas, F., Miralles, A., Liu, H., Zhang, Z.: The number of spanning trees of an infinite family of outerplanar, small-world and self-similar graphs. Phys. A 392(12), 2803–2806 (2013)

    Article  MathSciNet  Google Scholar 

  6. Chang, J.-H.: An embedding of multiple edge-disjoint Hamiltonian cycles on enhanced pyramid graphs. J. Inform. Process. Syst. 7(1), 75–84 (2011)

    Article  Google Scholar 

  7. Cao, F., Hsu, D.F.: Fault-tolerance properties of pyramid networks. IEEE Trans. Comput. 48, 88–93 (1999)

    Article  MathSciNet  Google Scholar 

  8. Dall’Asta, L., Alvarez-Hamelin, J.I., Barrat, A., Vázquez, A., Vespignani, A.: Exploring networks with traceroute-like probes: theory and simulations. Theoret. Comput. Sci. 355(1), 6–24 (2006)

    Article  MathSciNet  Google Scholar 

  9. Francesc, C., Alicia, M.: Vertex labeling and routing in self-similar outerplanar unclustered graphs modeling complex networks. J. Phys. A: Math. Theor. 42(42), 425001 (2009)

    Article  MathSciNet  Google Scholar 

  10. Foucaud, F., Kao, S., Klasing, R., Miller, M., Ryan, J.: Monitoring the edges of a graph using distances. Discrete Appl. Math. 319, 424–438 (2022)

    Article  MathSciNet  Google Scholar 

  11. Geetha, J., Somasundaram, K.: Total coloring of generalized Sierpiński graphs. Australas. J. Combin. 63(1), 58–69 (2015)

    MathSciNet  Google Scholar 

  12. Govindan, R., Tangmunarunkit, H.: Heuristics for Internet map discovery. In: Proc. IEEE INFOCOM Conf. Comput. Commun. 19th Annu. Joint Conf. IEEE Comput. Commun. Soc., vol. 3. Tel Aviv, Israel, pp. 1371–1380 (2000)

  13. Hoseinyfarahabady, M.R., Sarbazi-azad, H.: The grid-pyramid: a generalized pyramid network. J. Supercomput. 37, 23–45 (2006)

    Article  Google Scholar 

  14. Juan, J.S.-T., Huang, C.-M.: On the strong distance problems of pyramid networks. Appl. Math. Comput. 195(1), 154–161 (2008)

    MathSciNet  Google Scholar 

  15. Jeng, J.-F., Sahni, S.: Image shrinking and expanding on a pyramid. IEEE Trans. Parallel Distrib. Syst. 4, 1291–1296 (1993)

    Article  Google Scholar 

  16. Miller, R., Stout, Q.: Data movement techniques for the pyramid computer. SIAM J. Comput. 16, 38–60 (1987)

    Article  MathSciNet  Google Scholar 

  17. Teguia, A.M., Godbole, A.P.: Sierpiński gasket graphs and some of their properties. Australas. J. Combin. 35, 181–192 (2006)

    MathSciNet  Google Scholar 

  18. Zhang, Z., Zhou, S., Su, Z., et al.: Random Sierpiński network with scale-free small-world and modular structure. Eur. Phys. J. B 65(1), 141–147 (2008)

    Article  Google Scholar 

Download references

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 61, 183–198 (2024). https://doi.org/10.1007/s00236-024-00453-z

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