Let $G$ be a simple, connected non bipartite graph and let $I_G$ be the edge ideal of $G$. In our previous work we showed that L. Lov\'{a}sz's theorem on ear decompositions of factor-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}\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 $\Ass (I^{k}_G)=\Ass (I^{k+i}_G)$ for all $i\geq 0$ and the smallest number $k$ such that the maximal ideal belongs to $ \Ass (I^{k}_G)$. We also give very simple upper bounds for $\astab(I_G)$ and $\dstab(I_G)$.