Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las?Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion-contraction relations, its specialisation to cellularly embedded graphs does not. Here  extend the Las Vergnas polynomial to graphs in pseudo-surfaces.  show that in this setting  can define deletion and contraction for embedded graphs consistently with the deletion and contraction of the underlying matroid perspective, thus yielding a version of the Las Vergnas polynomial with complete recursive definition. This also enables us to obtain a deeper understanding of the relationships among the Las Vergnas polynomial, the Bollobas-Riordan polynomial, and the Krushkal polynomial.  also take this opportunity to extend some of Las Vergnas' results on Eulerian circuits from graphs in surfaces of low genus to graphs in surfaces of arbitrary genus.
['Joanna A. Ellis-Monaghan', 'Iain Moffatt']
I am writing a research paper on a idea The Las Vergnas polynomial for embedded graphs
[{"from": "human", "value": "I am writing a research paper on a idea The Las Vergnas polynomial for embedded graphs"}, {"from": "gpt", "value": "Thats a unique idea.You should consider the following idea to enhance your idea so that it would impact uniquely: The Las Vergnas polynomial is an extension of the Tutte polynomial to cellularly embedded graphs. It was introduced by Michel Las?Vergnas in 1978 as special case of his Tutte polynomial of a morphism of matroids. While the general Tutte polynomial of a morphism of matroids has a complete set of deletion-contraction relations, its specialisation to cellularly embedded graphs does not. Here  extend the Las Vergnas polynomial to graphs in pseudo-surfaces.  show that in this setting  can define deletion and contraction for embedded graphs consistently with the deletion and contraction of the underlying matroid perspective, thus yielding a version of the Las Vergnas polynomial with complete recursive definition. This also enables us to obtain a deeper understanding of the relationships among the Las Vergnas polynomial, the Bollobas-Riordan polynomial, and the Krushkal polynomial.  also take this opportunity to extend some of Las Vergnas' results on Eulerian circuits from graphs in surfaces of low genus to graphs in surfaces of arbitrary genus."}]