  We present bijections for the planar cases of two counting formulas on maps
that arise from the KP hierarchy (Goulden-Jackson and Carrell-Chapuy formulas),
relying on a "cut-and-slide" operation. This is the first time a bijective
proof is given for quadratic map-counting formulas derived from the KP
hierarchy. Up to now, only the linear one-faced case was known (Harer-Zagier
recurrence and Chapuy-F\'eray-Fusy bijection). As far as we know, this
bijection is new and not equivalent to any of the well-known bijections between
planar maps and tree-like objects.
