The $Δ$-edge stability number ${\rm es}_Δ(G)$ of a graph $G$ is the minimum number of edges of $G$ whose removal results in a subgraph $H$ with $Δ(H) = Δ(G)-1$. Sets whose removal results in a subgraph with smaller maximum degree are called mitigating sets. It is proved that there always exists a mitigating set which induces a disjoint union of paths of order $2$ or $3$. Minimum mitigating sets which induce matchings are characterized. It is proved that to obtain an upper bound of the form ${\rm es}_Δ(G) \leq c |V(G)|$ for an arbitrary graph $G$ of given maximum degree $Δ$, where $c$ is a given constant, it suffices to prove the bound for $Δ$-regular graphs. Sharp upper bounds of this form are derived for regular graphs. It is proved that if $Δ(G) \geq\frac{|V(G)|-2}{3}$ or the induced subgraph on maximum degree vertices has a $Δ(G)$-edge coloring, then ${\rm es}_Δ(G) \le \lceil |V(G)|/2\rceil$.