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New topological operations are introduced in order to recover the generalized Dyck equations presented by D. Arquès et al. in another way for the generating functions of maps and colored maps, by decomposing maps topologically and bijectively. By repeatedly applying the operations which made it possible to reveal the generalized Dyck equations for the successive transformed maps, one-to-one correspondences...
A simple quadrangulation of the sphere is a finite simple graph embedded on the sphere such that every face is bounded by a walk of 4 edges. We consider the following classes of simple quadrangulations: arbitrary, minimum degree 3, 3-connected, and 3-connected without non-facial 4-cycles. In each case, we show how the class can be generated by starting with some basic graphs in the class and applying...
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