Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: Given a plane graph G i.e., a planar graph with a fixed planar embedding and a simple cycle C in G whose vertices are mapped to a convex polygon,  consider the question whether this drawing can be extended to a planar straight-line drawing of G.  characterize when this is possible in terms of simple necessary conditions, which  prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, it can be computed in the same running time.
['Tamara Mchedlidze', 'Martin Nöllenburg', 'Ignaz Rutter']
I am writing a research paper on a idea Drawing Planar Graphs with a Prescribed Inner Face
[{"from": "human", "value": "I am writing a research paper on a idea Drawing Planar Graphs with a Prescribed Inner Face"}, {"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 plane graph G i.e., a planar graph with a fixed planar embedding and a simple cycle C in G whose vertices are mapped to a convex polygon,  consider the question whether this drawing can be extended to a planar straight-line drawing of G.  characterize when this is possible in terms of simple necessary conditions, which  prove to be sufficient. This also leads to a linear-time testing algorithm. If a drawing extension exists, it can be computed in the same running time."}]