Distance-edge-monitoring numbers of some related pseudo wheel networks

Abstract

For a set M of vertices and an edge e of a graph G, let $P_G(M,e)$ be the set of the pair $(x,y)$ with a vertex x of M and a vertex y of $V\left(G\right)$ such that $d_G(x,y)\ne d_{G-e}(x,y)$. For a vertex x, let $\mathrm{EM}\left(x\right)$ be the edge set e such that there exists a vertex v in G with $(x,v)\in P\left(\right\{x\},e)$. A set M of vertices of a graph G is distance-edge-monitoring set if every edge e of G is monitored by some vertex $v\in M$, that is, for any $e\in E\left(G\right)$, we have $P_G(M,e)\ne \mathrm{\varnothing }$. The distance-edge-monitoring number of a graph G, denoted by $\mathrm{dem}\left(G\right)$, is defined as the smallest size of distance-edge-monitoring sets of G. In this paper, we study the distance edge monitoring number of pseudo wheel graphs, that is, some variants of wheel graph.

KEYWORDS:

• Distance
• distance-edge-monitoring set
• pseudo wheel graph

AMS SUBJECT CLASSIFICATIONS 2020:

• 05D10
• 05D40
• 05C80

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

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