In mathematics, a Tutte–Grothendieck (TG) invariant is a type of graph invariant that satisfies a generalized deletion–contraction formula. Any evaluation of the Tutte polynomial would be an example of a TG invariant.
Definition A graph function f is TG-invariant if:
Above G / e denotes edge contraction whereas G \ e denotes deletion. The numbers c, x, y, a, b are parameters.
Generalization to matroids The matroid function f is TG if:
It can be shown that f is given by:
where is the value takes on coloops, is the value takes on loops, is the edge set of ; is the rank function; and
is the generalization of the Tutte polynomial to matroids.
Grothendieck group The invariant is named after Alexander Grothendieck because of a similar construction of the Grothendieck group used in the Riemann–Roch theorem. For more details see:
References