According to Grothendieck, the depth of a ring is a measure for how good is its structure. Let R be a standard graded algebra over a field and I a graded ideal. By a classical result of Brodmann, there exists a constant c such that depth R/I^t = depth R/I^c for all t \ge c. Let dstab(I) denote the minimal number c with this property. It is extremely hard to determine dstab(I) in general. This problem can be solved if I is the edge ideal of a graph. In this talk we will give characterizations of dstab(I) in terms of ear decompositions of the graph, which are based on earlier works of Lovasz on matching-critical and matching covered graphs.