Monitoring the edges of a graph using distances with given girth

A set M of vertices of a graph G is a distance-edge-monitoring set if for every edge $e\in G$, there is a vertex $x\in M$ and a vertex $y\in G$ such that e belongs to all shortest paths between x and y. We denote by $\mathrm{dem}\left(G\right)$ the smallest size of such a set in G. In this paper, we prove that $\mathrm{dem}\left(G\right)\le n-\lfloor g\left(G\right)/2\rfloor$ for any connected graph G, which is not a tree, of order n, where $g\left(G\right)$ is the length of a shortest cycle in G, and give the graphs with $\mathrm{dem}\left(G\right)=n-\lfloor g\left(G\right)/2\rfloor$. We also obtain that $\left|V\right(G\left)\right|\ge k+\lfloor g\left(G\right)/2\rfloor$ for every connected graph G with $\mathrm{dem}\left(G\right)=k$ and $g\left(G\right)=g$. Furthermore, the lower bound holds if and only if $g=3$ and $k=n-1$ or $g=4$ and $k=2$. We prove that $\mathrm{dem}\left(G\right)\le 2n/5$ for $g\left(G\right)\ge 5$.