On a conjecture of Gross, Mansour and Tucker

Partial duality is a of ribbon graphs relative to subset their edges generalizing the classical Euler–Poincaré duality. This operation often changes genus. Recently J.L. Gross, T. Mansour, and T.W. Tucker formulated conjecture that for any graph different from plane trees partial duals, there which does change A family counterexamples was found by Qi Yan Xian’an Jin. In this note we prove essentially these are only counterexamples.