If one doesn't want to introduce additional variables to the ILP of a TSP instance, one has to add exponentially many so-called subtour-elimination constraints; in practical calculations subtour-elimination constraints are added lazily, i.e. as subtours are encountered.  
The number of subtour elimination constraints that are added after every iteration is $O(n)$.  


>**Question:**   
>
>if after an iteration in calculating the optimal tour via a sequence of ILPs $k$ subtours are encountered:  
wouldn't it be more efficient to  
>- replace the $k$ subtour elimination constraints  
>- with the *single* "subtour gluing constraint"  
> 
>that demands that summing over the variables that correspond to edges that are adjacent to two vertices from different subtours from the current set of subtours?  

**Addendum:**   

to clarify what kind of constraint is proposed I denote by $S_i(V_i,E_i)$ the subgraph induced by the $n_i$ vertices $V_i$ of the $i$-th subtour of the current iteration;  
I further denote by $|S_i|$ the number of that subgraph's edges that are in the optimal solution of the next iteration.  

With the above notation  
- the classical subtour elimination constraints would be $|S_h|\,\lt\, n_h,\ h=1,\,\dots,\,k$ and 
- the proposed subtour gluing constraint would be $$\sum\limits_{\lbrace i,j\rbrace: e_{ij}\ \in\ E(G)\,\setminus\,\bigcup\limits_{h=1}^k E_h}x_{ij}\quad\ge\quad k$$