Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  introduce and study the Doubly Balanced Connected graph Partitioning (DBCP) problem: Let  G =( V, E ) be a connected graph with a ight (supply/demand) function  p : V →{−1, +1} satisfying  p ( V )=Σ  j ∈ v    p ( j )=0. The objective is to partition  G  into ( V  1 ,  V  2 ) such that  G [ V  1 ] and  G [ V  2 ] are connected, | p ( V  1 )|, | p ( V  2 )|≤ c p  , and max [EQUATION], for some constants  c p   and  c s  . When  G  is 2-connected,  show that a solution with  c p  =1 and  c s  =3 always exists and can be found in polynomial time. Moreover, when  G  is 3-connected,  show that there is always a 'perfect' solution (a partition with  p ( V  1 )= p ( V  2 )=0 and | V  1 |=| V  2 |, if | V |≡0(mod 4)), and it can be found in polynomial time. Our techniques can be extended, with similar results, to the case in which the ights are arbitrary (not necessarily ±1), and to the case that  p ( V )≠0 and the excess supply/demand should be split evenly. They also apply to the problem of partitioning a graph with two types of nodes into two large connected subgraphs that preserve approximately the proportion of the two types.
['Saleh Soltan', 'Mihalis Yannakakis', 'Gil Zussman']
I am writing a research paper on a idea Doubly balanced connected graph partitioning
[{"from": "human", "value": "I am writing a research paper on a idea Doubly balanced connected graph partitioning"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  introduce and study the Doubly Balanced Connected graph Partitioning (DBCP) problem: Let  G =( V, E ) be a connected graph with a ight (supply/demand) function  p : V \u2192{\u22121, +1} satisfying  p ( V )=\u03a3  j \u2208 v    p ( j )=0. The objective is to partition  G  into ( V  1 ,  V  2 ) such that  G [ V  1 ] and  G [ V  2 ] are connected, | p ( V  1 )|, | p ( V  2 )|\u2264 c p  , and max [EQUATION], for some constants  c p   and  c s  . When  G  is 2-connected,  show that a solution with  c p  =1 and  c s  =3 always exists and can be found in polynomial time. Moreover, when  G  is 3-connected,  show that there is always a 'perfect' solution (a partition with  p ( V  1 )= p ( V  2 )=0 and | V  1 |=| V  2 |, if | V |\u22610(mod 4)), and it can be found in polynomial time. Our techniques can be extended, with similar results, to the case in which the ights are arbitrary (not necessarily \u00b11), and to the case that  p ( V )\u22600 and the excess supply/demand should be split evenly. They also apply to the problem of partitioning a graph with two types of nodes into two large connected subgraphs that preserve approximately the proportion of the two types."}]