On terminal forms for topological polynomials for ribbon graphs: The N-petal flower

The Bollobas-Riordan polynomial [B. Bollobas, O. Riordan, A polynomial of graphs on surfaces, Math. Ann. 323 (2002) 81-96] extends the Tutte polynomial and its contraction/deletion rule for ordinary graphs to ribbon graphs. Given a ribbon graph G, the related polynomial should be computable from the knowledge of the terminal forms of G namely specific induced graphs for which the contraction/deletion procedure becomes more involved. We consider some classes of terminal forms as rosette ribbon graphs with N>=1 petals and solve their associate Bollobas-Riordan polynomial. This work therefore enlarges the list of terminal forms for ribbon graphs for which the Bollobas-Riordan polynomial could be directly deduced.