Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given an unighted tree $T=(V,E)$ with terminals $K \subset V$,  show how to obtain a $2$-quality vertex flow and cut sparsifier $H$ with $V_H = K$.  prove that our result is essentially tight by providing a $2-o(1)$ lor-bound on the quality of any cut sparsifier for stars. In addition  give improved results for quasi-bipartite graphs. First,  show how to obtain a $2$-quality flow sparsifier with $V_H = K$ for such graphs.  then consider the other extreme and construct exact sparsifiers of size $O(2^{k})$, when the input graph is unighted.
['Gramoz Goranci', 'Harald Räcke']
I am writing a research paper on a idea Vertex Sparsification in Trees
[{"from": "human", "value": "I am writing a research paper on a idea Vertex Sparsification in Trees"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given an unighted tree $T=(V,E)$ with terminals $K \\subset V$,  show how to obtain a $2$-quality vertex flow and cut sparsifier $H$ with $V_H = K$.  prove that our result is essentially tight by providing a $2-o(1)$ lor-bound on the quality of any cut sparsifier for stars. In addition  give improved results for quasi-bipartite graphs. First,  show how to obtain a $2$-quality flow sparsifier with $V_H = K$ for such graphs.  then consider the other extreme and construct exact sparsifiers of size $O(2^{k})$, when the input graph is unighted."}]