On Ribbon Graphs that Admit a Partial Dual of Euler Genus at Most Two

Moffatt in his paper Excluded minors and the ribbon graphs of knots (Journal of Graph Theory, 2016), conjectures that every ribbon graph minor-closed family can be characterized by a finite set of excluded ribbon graph minors. He supports this conjecture in several papers, particularly in Ribbon graph minors and low-genus partial duals (Annals of Combinatorics, 2016), by giving a finite list of excluded minors that characterizes the class of ribbon graphs with a partial dual of Euler genus at most one. In this paper, we give a finite list of excluded minors that characterizes ribbon graphs with a partial dual of Euler genus at most two, subject to the condition that any bouquet related by partial duality to the ribbon graph satisfies that the intersection graph of the induced subgraph of its non-orientable loops and the complement of the intersection graph of the induced subgraph of its orientable loops are both $3$-cycle free.