  Let $G$ be a simple, connected non bipartite graph and let $I_G$ be the edge
idealof $G$. In our previous work we showed that L. Lov\'asz's theorem on ear
decompositions offactor-critical graphs and the canonical decomposition of a
graph given by Edmonds and Gallai are basic tools for the irreducible
decomposition of $I^{k}_G$. In this paper we use some tools from graph theory,
mainly Withney's theorem on ear decompositions of 2-edge connected graphs in
order to introduce a new method to make a graph factor-critical. We can
describe the set $\cup_ {k=1}^{\infty}{\rm Ass} (I^{k}_G)$ in terms of some
subsets of $G$. We give explicit formulas for the numbers astab$(I_G)$ and
dstab$(I_G)$, which are, respectively, the smallest number $k$ such that ${\rm
Ass} (I^{k}_G)= {\rm Ass} (I^{k+i}_G)$ for all $i\geq 0$ and the smallest
number $k$ such that the maximal ideal belongs to ${\rm Ass}(I^{k}_G)$. We also
give very simple upper bounds for astab$(I_G)$ and dstab$(I_G)$.
