For an arbitrary invariant ρ(G) of a graph G the ρ-vertex stability number vsρ(G) (ρ-edge stability number esρ(G)) is the minimum number of vertices (edges) whose removal results in a graph H⊆G with ρ(H)≠ρ(G). If such a vertex set (edge set) does not exist, then we set vsρ(G)=∞ (esρ(G)=∞).In the first part of this paper we give some general results for the ρ-vertex stability number. We prove among others a result which implies the well-known Theorem of Gallai on independence and vertex covering of graphs. In the second part we focus on the invariant chromatic number χ(G) and determine vsχ(G) for several classes of graphs. Moreover, we consider the relationship between esχ(G) and vsχ(G) and prove that their difference and their ratio may get arbitrarily large.