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