  We introduce a new graph-theoretic concept in the area of network monitoring.
A set $M$ of vertices of a graph $G$ is a \emph{distance-edge-monitoring set}
if for every edge $e$ of $G$, there is a vertex $x$ of $M$ and a vertex $y$ of
$G$ such that $e$ belongs to all shortest paths between $x$ and $y$. We denote
by $dem(G)$ the smallest size of such a set in $G$. 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 initiate the study of this new concept. We show that for a
nontrivial connected graph $G$ of order $n$, $1\leq dem(G)\leq n-1$ with
$dem(G)=1$ if and only if $G$ is a tree, and $dem(G)=n-1$ if and only if it is
a complete graph. We compute the exact value of $dem$ for grids, hypercubes,
and complete bipartite graphs.
  Then, we relate $dem$ to other standard graph parameters. We show that
$demG)$ is lower-bounded by the arboricity of the graph, and upper-bounded by
its vertex cover number. It is also upper-bounded by twice its feedback edge
set number. Moreover, we characterize connected graphs $G$ with $dem(G)=2$.
  Then, we show that determining $dem(G)$ for an input graph $G$ is an
NP-complete problem, even for apex graphs. There exists a polynomial-time
logarithmic-factor approximation algorithm, however it is NP-hard to compute an
asymptotically better approximation, even for bipartite graphs of small
diameter and for bipartite subcubic graphs. For such instances, the problem is
also unlikey to be fixed parameter tractable when parameterized by the solution
size.
