Counting two types of quadrangulations: Rooted near quadrangulations on the disc and nonseparable outerplanar quadrangulations

In this paper, we provide functional equations satisfied by the generating functions for enumerating rooted near quadrangulations on the disc and rootednonseparable outerplanar quadrangulations dependent on the edgenumber and the valency of the root-face respectively. Furthermore, we present a summation-free formula for rooted nonseparableouterplanar quadrangulations and an explicit formula for rooted nearquadrangulations on the disc by employing Lagrangian inversion basedon the cubic enunfunctions. As consequences, the number of rootedHamiltonian planar quadrangulations with even order and rooted (4,3)-regular Halin map are extracted more directly and more simply.