Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally,  say that R is column planar in G if  can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity.  prove near tight bounds for column planar subsets of trees: any tree on n vertices contains a column planar set of size at least 14n/17 and for any e>0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at mosti¾?5/6+en.#R##N##R##N# also consider a relaxation of simultaneous geometric embedding SGE, which  call partial SGE PSGE. A PSGE of two graphs G 1 and G 2 allows some of their vertices to map to two different points in the plane.  show how to use column planar subsets to construct k-PSGEs in which k vertices are still mapped to the same point. In particular,  show that any two trees on n vertices admit an 11n/17-PSGE, two outerpaths admit an n/4-PSGE, and an outerpath and a tree admit a 11n/34-PSGE.
['William S. Evans', 'Vincent Kusters', 'Maria Saumell', 'Bettina Speckmann']
I am writing a research paper on a idea Column Planarity and Partial Simultaneous Geometric Embedding
[{"from": "human", "value": "I am writing a research paper on a idea Column Planarity and Partial Simultaneous Geometric Embedding"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely:  introduce the notion of column planarity of a subset R of the vertices of a graph G. Informally,  say that R is column planar in G if  can assign x-coordinates to the vertices in R such that any assignment of y-coordinates to them produces a partial embedding that can be completed to a plane straight-line drawing of G. Column planarity is both a relaxation and a strengthening of unlabeled level planarity.  prove near tight bounds for column planar subsets of trees: any tree on n vertices contains a column planar set of size at least 14n/17 and for any e>0 and any sufficiently large n, there exists an n-vertex tree in which every column planar subset has size at mosti\u00be?5/6+en.#R##N##R##N# also consider a relaxation of simultaneous geometric embedding SGE, which  call partial SGE PSGE. A PSGE of two graphs G 1 and G 2 allows some of their vertices to map to two different points in the plane.  show how to use column planar subsets to construct k-PSGEs in which k vertices are still mapped to the same point. In particular,  show that any two trees on n vertices admit an 11n/17-PSGE, two outerpaths admit an n/4-PSGE, and an outerpath and a tree admit a 11n/34-PSGE."}]