The multiple origin/destination connection Problem is as follows
Let $G=(V,E)$ be an undirected graph, and let $c_e$ be a given cost for each $e\in E$. Let $(s_k, t_k)$ for $k=1,...,p\ $ be a given list of origin/destination pairs with $s_k, t_k\in V$ for each k. We seek to choose a set of edges Q ⊆ E of minimum cost such that for each O-D pair k, $s_k$ and $t_k$ will remain connected after removal of any set of m-1 edges of Q.
Binary variables $x_e$ are introduced for $e\in E$, where $x_e=1$ if and only if $e\in Q$. The objective is hence to minimize $\sum c_ex_e$.
Formulations by adding additional variables is possible. I just wonder if it is possible to form the problem as an integer problem without introducing any extra variable, i.e., using only the variables $x_e$? If yes, then how is it formed?
Yes, it suffices to impose constraints of the form $\sum_{E \setminus S} x_e \ge 1$, where $S \subset E$, $|S|=m-1$, and removal of $S$ separates $s_k$ from $t_k$ for some $k$. This constraint forces at least one edge outside of $S$ to appear in $Q$. There are typically exponentially many such constraints, but you can generate them dynamically if they are violated.