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I am writing the terms "Explicit set of Tools" or the " Mathematical Tools" ; which means the mathematical machinary needed to enumerate face vectors associated with polyhedron, to calculate the combinatorial types.
Joseph Malkevitch's own AMS monthly column "Euler's Polyhedral Formula: PartII" is also a nice elaboration in this respect; but naming the explicit set of tools will be helpful.
Indeed, Joseph Malkevitch's exposition "Euler's Polyhedral Formula: Part II" is a nice article in AMS archive. Comments on different directions of attack on the problem will be appreciated.