  Foucaud et al. recently introduced and initiated the study of a new
graph-theoretic concept in the area of network monitoring. Given a graph
$G=(V(G), E(G))$, a set $M \subseteq V(G)$ is a distance-edge-monitoring set if
for every edge $e \in E(G)$, there is a vertex $x \in M$ and a vertex $y \in
V(G)$ such that the edge $e$ belongs to all shortest paths between $x$ and $y$.
The smallest size of such a set in $G$ is denoted by $\operatorname{dem}(G)$.
Denoted by $G-e$ (resp. $G \backslash u$) the subgraph of $G$ obtained by
removing the edge $e$ from $G$ (resp. a vertex $u$ together with all its
incident edges from $G$). In this paper, we first show that
$\operatorname{dem}(G-e)- \operatorname{dem}(G)\leq 2$ for any graph $G$ and
edge $e \in E(G)$. Moreover, the bound is sharp. Next, we construct two graphs
$G$ and $H$ to show that $\operatorname{dem}(G)-\operatorname{dem}(G\setminus
u)$ and $\operatorname{dem}(H\setminus v)-\operatorname{dem}(H)$ can be
arbitrarily large, where $u \in V(G)$ and $v \in V(H)$. We also study the
relation between $\operatorname{dem}(H)$ and $\operatorname{dem}(G)$, where $H$
is a subgraph of $G$. In the end, we give an algorithm to judge whether the
distance-edge monitoring set still remain in the resulting graph when any edge
of the graph $G$ is deleted.
