  Foucaud et al. [Discrete Appl. Math. 319 (2022), 424-438] recently introduced
and initiated the study of a new graph-theoretic concept in the area of network
monitoring. For a set $M$ of vertices and an edge $e$ of a 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 $d_G(x, y)\neq d_{G-e}(x, y)$. For a vertex $x$, let $EM(x)$
be the set of edges $e$ such that there exists a vertex $v$ in $G$ with $(x, v)
\in P(\{x\}, e)$. 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 of $M$, that is, the set $P(M, e)$ is nonempty. The
distance-edge-monitoring number of a graph $G$, denoted by $dem(G)$, is defined
as the smallest size of distance-edge-monitoring sets of $G$. The vertices of
$M$ represent distance probes in a network modeled by $G$; when the edge $e$
fails, the distance from $x$ to $y$ increases, and thus we are able to detect
the failure. It turns out that not only we can detect it, but we can even
correctly locate the failing edge. In this paper, we continue the study of
\emph{distance-edge-monitoring sets}. In particular, we give upper and lower
bounds of $P(M,e)$, $EM(x)$, $dem(G)$, respectively, and extremal graphs
attaining the bounds are characterized. We also characterize the graphs with
$dem(G)=3$.
