Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It was proved ([5], [6]) that ifG is ann-vertex-connected graph then for any vertex sequencev1, ...,vn ≠V(G) and for any sequence of positive integersk1, ...,kn such thatk1+...+kn=|V(G)|, there exists ann-partition ofV(G) such that this partition separates the verticesv1, ...,v(n), and the class of the partition containingvi induces a connected subgraph consisting ofki vertices, fori=1, 2, ...,n. Now fix the integersk1, ...,kn.   study what can  say about the vertex-connectivity ofG if there exists such a partition ofV(G) for any sequence of verticesv1, ...,vn ≠V(G).  find some interesting cases when the existence of such partitions implies then-vertex-connectivity ofG, in the other cases  give sharp lor bounds for the vertex-connectivity ofG.
['Ervin Győri']
I am writing a research paper on a idea Partition conditions and vertex-connectivity of graphs
[{"from": "human", "value": "I am writing a research paper on a idea Partition conditions and vertex-connectivity of graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: It was proved ([5], [6]) that ifG is ann-vertex-connected graph then for any vertex sequencev1, ...,vn \u2260V(G) and for any sequence of positive integersk1, ...,kn such thatk1+...+kn=|V(G)|, there exists ann-partition ofV(G) such that this partition separates the verticesv1, ...,v(n), and the class of the partition containingvi induces a connected subgraph consisting ofki vertices, fori=1, 2, ...,n. Now fix the integersk1, ...,kn.   study what can  say about the vertex-connectivity ofG if there exists such a partition ofV(G) for any sequence of verticesv1, ...,vn \u2260V(G).  find some interesting cases when the existence of such partitions implies then-vertex-connectivity ofG, in the other cases  give sharp lor bounds for the vertex-connectivity ofG."}]