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We unify and extend previous bijections on plane quadrangulations tobipartite and quasibipartite plane maps. Starting from a bipartite plane mapwith a distinguished edge and two distinguished corners (in the same face or intwo different faces), we build a new plane map with a distinguished vertex andtwo distinguished half-edges directed toward the vertex. The faces of the newmap have the same degree as those of the original map, except at the locationsof the distinguished corners, where each receives an extra degree. The ideabehind this bijection is to build a path from the distinguished elements, slitthe map along it, and sew back after sliding by one unit, thus mildly modifyingthe structure of the map at the extremities of the sliding path. This bijectionprovides a sampling algorithm for uniform maps with prescribed face degrees andallow to recover Tutte's famous counting formula for bipartite andquasibipartite plane maps.  In addition, we explain how to decompose the previous bijection into two moreelementary ones, which each transfer a degree from one face of the map toanother face. In particular, these transfer bijections are simpler tomanipulate than the previous one and this point of view simplifies the proofs.

Slit-slide-sew bijections for bipartite and quasibipartite plane maps