How does the Vertex Cover algorithm by Chen et al find its tuples?
The beginning of Section 4 states: 


  A tuple, a good pair, or a vertex of degree at least seven, will be referred to by the word structure. The algorithm will
  maintain a list of structures $\mathcal{T}$, and then it will pick a structure and process it. ... We will assume that the algorithm implicitly
  updates the structures in $\mathcal{T}$ and their priorities after each operation.


It seems that you shouldn't really think of $\mathcal{T}$ as a parameter, but as a dynamically updated list. As for intuition, I suggest that you read the full paper. Once you understand the proof, you should be able to come up with some sort of intuition.