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. Twisted duality is a further generalization from combining partial Petrials with partial duals. Ribbon graphs are a new equivalent form of the old cellularly embedded graphs. In this paper, we first characterize regular partial duals of a ribbon graph by using spanning quasi-tree and its related nesting 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 solves a problem raised by Ellis-Monaghan and Moffatt (2012) [10].

细胞嵌入图的几何对偶是图论中的基本概念，并且也出现在数学的许多其他分支中。部分对偶是基本的概括，可以通过仅对单元格嵌入图的边缘的子集形成几何对偶来获得。扭曲对偶性是将部分Petrial与部分对偶结合在一起的进一步推广。功能区图是旧的蜂窝嵌入图的新等效形式。在本文中，我们首先通过使用生成树及其相关的嵌套集来表征带状图的规则部分对偶。然后，通过使用生成树和相关的伴随集概念，为任何欧拉带状图表征棋盘色有色Petrial。