Set of the vertex sets to make connected graph into disjoint sets of vertices? Suppose a non-directed graph G with vertices V and paths P. What is the name for the vertex sets to make break the graph by removal of some vertices?
 A: I list below the relevant things in Mathematics. In comparison, the computing perspectives and visualisation more here. 
Theorems and Lemmas

  
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*Planar separator theorem
  
*Tutte theorem related to edge cuts
  
*Minimal vertex separator lemma

  
  
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*Menger's theorem has vertex-connectivity version and edge-connectivity version
  

Examples

Dominating set suggested in the comment by dREaM, from dominating sets to set coverings in L-reductions here, Wikipedia about dominating set example below.

Cut set and vertex separator suggested in the comment by JMoravitz here and here, respectively. The examples are from the vertex separator Wikipedia article.

Vertex cuts and edge cuts suggested in the comment by M.U. related to infinite graph theory by Dunwoody and Krön, more in the paper "Vertex Cuts" by Dunwoody, Krön, also possible to apply to finite graphs according to M.U. 
Graph separators, graph bifurcators, graph boundaries -- loosely speaking result into two separate subgraphs after removal of some vertices or some edges. (On page 12 of Graph Separators, with Applications by Arnold L. Rosenberg, et all).

References

  
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*Vertex cuts publication by Dunwoody, Krön as suggested by M.U.
  
*Book "Groups acting on Graphs" by Dicks, Dunwoody suggested by M.U.
  
*A Separator Theorem for Planar Graphs and A Separator Theorem for Graphs with an Excluded Minor and its Applications
  
*Graph-connectivity
  
*Cactus presentation for mincuts in undirected, unweighted graphs

