  Given a non-trivial graph $G$, the minimum cardinality of a set of edges $F$
in $G$ such that $\chi'(G \setminus F)<\chi'(G)$ is called the chromatic edge
stability index of $G$, denoted by $es_{\chi'}(G)$, and such a (smallest) set
$F$ is called a (minimum) mitigating set. While $1\le es_{\chi'}(G)\le \lfloor
n/2\rfloor$ holds for any graph $G$, we investigate the graphs with extremal
and near-extremal values of $es_{\chi'}(G)$. The graphs $G$ with
$es_{\chi'}(G)=\lfloor n/2\rfloor$ are classified, and the graphs $G$ with
$es_{\chi'}(G)=\lfloor n/2\rfloor-1$ and $\chi'(G)=\Delta(G)+1$ 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
$es_{\chi'}(G)=1$. We also prove that every minimum mitigating set of an
$r$-regular graph $G$, where $r\ne 4$, with $es_{\chi'}(G)=2$ 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 $es_{\chi'}(G)\in\{1,2,\lfloor n/2\rfloor-1,\lfloor
n/2\rfloor\}$, and for bipartite graphs.
