Partial Petrial polynomials of ribbon graphs equal sums over bouquets, and a signed modification is a 4-invariant for signed simple graphs.
Cheng, Partial-dual genus polynomial of graphs, European Journal of Combinatorics , 130 (2025): 104221
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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.