Given a non-trivial graph G, the minimum cardinality of a set of edges F in G such that χ′(G∖ F)< χ′(G) is called the chromatic edge stability index of G, denoted by e s χ′(G), and such a (smallest) set F is called a (minimum) mitigating set. While 1≤ e s χ′(G)≤⌊ n/2⌋ holds for any graph G, we investigate the graphs with extremal and near-extremal values of e s χ′(G). The graphs G with e s χ′(G)=⌊ n/2⌋ are classified, and the graphs G with e s χ′(G)=⌊ n/2⌋− 1 and χ′(G)= Δ (G)+ 1 are characterized. We establish that the odd cycles and K 2 are exactly the regular connected graphs with the chromatic edge stability index 1; on the other hand, we prove that it is NP-hard to verify whether a graph G has e s χ′(G)= 1. We also prove that every minimum mitigating set of an r-regular graph G, where r≠ 4, with e s χ′(G)= 2 is a matching. Furthermore, we propose a conjecture that for every graph G there exists a minimum mitigating set, which is a matching, and prove that the conjecture holds for graphs G with e s χ′(G)∈{1, 2,⌊ n/2⌋− 1,⌊ n/2⌋}, and for bipartite graphs.