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T. K\'{a}lm\'{a}n (A version of Tutte's polynomial for hypergraphs, Adv.Math. 244 (2013) 823-873.) introduced the interior and exterior polynomialswhich are generalizations of the Tutte polynomial $T(x,y)$ on plane points$(1/x,1)$ and $(1,1/y)$ to hypergraphs. The two polynomials are defined under afixed ordering of hyperedges, and are proved to be independent of the orderingusing techniques of polytopes. In this paper, similar to the Tutte's originalproof we provide a direct and elementary proof for the well-definedness of theinterior and exterior polynomials of hypergraphs.

A direct and elementary proof of the well-definedness of the interior and exterior polynomials of hypergraphs