The main advantage of the edge metric basis in the detection problem is that it identifies every distinct pair of edges in a simple connected graph [Formula: see text] by placing detectors at selected vertices. However, if one detector fails, the entire system breaks down. To address this, we define the [Formula: see text]-edge metric generator (or [Formula: see text]-edge resolving set). A [Formula: see text]-edge metric generator [Formula: see text] is a nonempty set of vertices such that for every pair of distinct edges in [Formula: see text], there are at least [Formula: see text] vertices in [Formula: see text] that uniquely identify them. The [Formula: see text]-edge metric generator of minimum size is called the [Formula: see text]-edge metric basis. Its size is the [Formula: see text]-edge metric dimension of [Formula: see text], written as [Formula: see text]. If [Formula: see text] is the largest integer for which a [Formula: see text]-edge metric basis exists in a connected graph [Formula: see text], then [Formula: see text] is a [Formula: see text]-edge metric dimensional graph. We characterize both [Formula: see text]-edge and [Formula: see text]-edge metric dimensional graphs and provide a necessary and sufficient condition to compute the maximum possible [Formula: see text]. We also prove that calculating the [Formula: see text]-edge metric dimension of a graph is an NP-complete problem.