  For an arbitrary invariant $\rho (G)$ of a graph $G$, the $\rho-$vertex
stability number $vs_{\rho}(G)$ is the minimum number of vertices of $G$ whose
removal results in a graph $H\subseteq G$ with $\rho (H)\neq \rho (G)$ or with
$E(H)=\varnothing$. In this paper, first we give some general lower and upper
bounds for the $\rho$-vertex stability number, and then study the edge
chromatic stability number of graphs, $vs_{\chi^{\prime}}(G)$, where
$\chi^{\prime}=\chi^{\prime}(G)$ is edge chromatic number (chromatic index) of
$G$. We prove some general results for this parameter and determine
$vs_{\chi^{\prime}}(G)$ for specific classes of graphs.
