Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: For t>0 and g≥0, a vertex-ighted graph of total ight W is (t,g)-trimmable if it contains a vertex-induced subgraph of total ight at least (1−1/t)W and with no simple path of more than g edges. A family of graphs is trimmable if for every constant t>0, there is a constant g≥0 such that every vertex-ighted graph in the family is (t,g)-trimmable.  show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar.  also show that every family of directed graphs of bounded layer bandwidth (a less restrictive condition than bounded directed bandwidth) is trimmable. As an application of these results,  derive polynomial-time approximation schemes for various forms of the problem of labeling a subset of given ighted point features with nonoverlapping sliding axes-parallel rectangular labels so as to maximize the total ight of the labeled features, provided that the ratios of label heights or the ratios of label lengths are bounded by a constant. This settles one of the last major open questions in the theory of map labeling.
['Thomas Erlebach', 'Torben Hagerup', 'Klaus Jansen', 'Moritz Minzlaff', 'Alexander Wolff']
I am writing a research paper on a idea Trimming of Graphs, with Application to Point Labeling
[{"from": "human", "value": "I am writing a research paper on a idea Trimming of Graphs, with Application to Point Labeling"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: For t>0 and g\u22650, a vertex-ighted graph of total ight W is (t,g)-trimmable if it contains a vertex-induced subgraph of total ight at least (1\u22121/t)W and with no simple path of more than g edges. A family of graphs is trimmable if for every constant t>0, there is a constant g\u22650 such that every vertex-ighted graph in the family is (t,g)-trimmable.  show that every family of graphs of bounded domino treewidth is trimmable. This implies that every family of graphs of bounded degree is trimmable if the graphs in the family have bounded treewidth or are planar.  also show that every family of directed graphs of bounded layer bandwidth (a less restrictive condition than bounded directed bandwidth) is trimmable. As an application of these results,  derive polynomial-time approximation schemes for various forms of the problem of labeling a subset of given ighted point features with nonoverlapping sliding axes-parallel rectangular labels so as to maximize the total ight of the labeled features, provided that the ratios of label heights or the ratios of label lengths are bounded by a constant. This settles one of the last major open questions in the theory of map labeling."}]