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The geometric dual of a cellularly embedded graph is a fundamental concept ingraph theory and also appears in many other branches of mathematics. Thepartial dual is an essential generalization which can be obtained by formingthe geometric dual with respect to only a subset of edges of a cellularlyembedded graph. The twisted dual is a further generalization by combining thepartial Petrial. Given a ribbon graph $G$, in this paper, we first characterizeregular partial duals of the ribbon graph $G$ by using spanning quasi-tree andits related shorter marking arrow sequence set. Then we characterizecheckerboard colourable partial Petrials for any Eulerian ribbon graph by usingspanning trees and a related notion of adjoint set. Finally we give a completecharacterization of all regular checkerboard colourable twisted duals of aribbon graph, which solve a problem raised by Ellis-Monaghan and Moffatt [T.Am. Math. Soc., 364(3) (2012), 1529-1569].

Characterization of regular checkerboard colourable twisted duals of ribbon graphs