Partial Petrial polynomials of ribbon graphs equal sums over bouquets, and a signed modification is a 4-invariant for signed simple graphs.
Beck, Cycle decomposition by transpositions, Journal of Combinatorial Theory, Series B , 23 (2) (1977): 198–207
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Partial Petrial Polynomials of Ribbon Graphs
Partial Petrial polynomials of ribbon graphs equal sums over bouquets, and a signed modification is a 4-invariant for signed simple graphs.