Monitoring-edge-geodetic numbers of radix triangular mesh and Sierpiński graphs

Abstract

Given a graph G and an edge $e\in E\left(G\right)$, let S be a vertex set of G. For any two vertices x, $y\in S$, if e belongs to all the shortest paths between x and y, then x and y can monitor the edge e. For each edge e of G, if there exists x and y in S such that x and y can monitor e, then the set S can be called a monitoring-edge-geodetic ($\mathrm{MEG}$ for short) set of G. The $\mathrm{MEG}$ number, denoted by $\mathrm{meg}\left(G\right)$, is the size of the smallest $\mathrm{MEG}$ set of G. In this paper, we obtain the exact values of the $\mathrm{MEG}$ numbers for radix triangular mesh networks, Sierpiński graphs, Sierpiński gasket graphs and Sierpiński generalized graphs.

KEYWORDS:

• Distance
• monitoring edge-geodetic set
• Sierpiński graph
• radix triangular mesh
• network

AMS SUBJECT CLASSIFICATIONS 2020:

• 05C12
• 05C35
• 05C82

Acknowledgments

We would like to thank the anonymous referees for a number of helpful comments and suggestions.

Disclosure statement

No potential conflict of interest was reported by the author(s).

Additional information

Funding

This work was supported by by the National Science Foundation of China (No. 12061059), the Qinghai Key Laboratory of Internet of Things Project (2017-ZJ-Y21).