Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a triangulated planar graph G on n vertices and an integer r r--division  of G with  few holes  is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called  holes ), and such that, for each region, the total number of vertices on these faces is O(√ r).  provide an algorithm for computing r--divisions with few holes in linear time. In fact, our algorithm computes a structure, called  decomposition tree , which represents a recursive decomposition of G that includes r--divisions for essentially all values of r. In particular, given an exponentially increasing sequence {vec r} = (r 1 ,r 2 ,...), our algorithm can produce a  recursive  {vec r}--division with few holes in linear time.      r--divisions with few holes have been used in efficient algorithms to compute shortest paths, minimum cuts, and maximum flows. Our linear-time algorithm improves upon the decomposition algorithm used in the state-of-the-art algorithm for minimum st--cut (Italiano, Nussbaum, Sankowski, and Wulff-Nilsen, STOC 2011), removing one of the bottlenecks in the overall running time of their algorithm (analogously for minimum cut in planar and bounded-genus graphs).
['Philip N. Klein', 'Shay Mozes', 'Christian Sommer']
I am writing a research paper on a idea Structured recursive separator decompositions for planar graphs in linear time
[{"from": "human", "value": "I am writing a research paper on a idea Structured recursive separator decompositions for planar graphs in linear time"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a triangulated planar graph G on n vertices and an integer r r--division  of G with  few holes  is a decomposition of G into O(n/r) regions of size at most r such that each region contains at most a constant number of faces that are not faces of G (also called  holes ), and such that, for each region, the total number of vertices on these faces is O(\u221a r).  provide an algorithm for computing r--divisions with few holes in linear time. In fact, our algorithm computes a structure, called  decomposition tree , which represents a recursive decomposition of G that includes r--divisions for essentially all values of r. In particular, given an exponentially increasing sequence {vec r} = (r 1 ,r 2 ,...), our algorithm can produce a  recursive  {vec r}--division with few holes in linear time.      r--divisions with few holes have been used in efficient algorithms to compute shortest paths, minimum cuts, and maximum flows. Our linear-time algorithm improves upon the decomposition algorithm used in the state-of-the-art algorithm for minimum st--cut (Italiano, Nussbaum, Sankowski, and Wulff-Nilsen, STOC 2011), removing one of the bottlenecks in the overall running time of their algorithm (analogously for minimum cut in planar and bounded-genus graphs)."}]