We extend a recent analysis of the $q$-states Potts model on an ensemble of random planar graphs with $pleqslant q$ allowed, equally weighted, spins on a connected boundary. In this paper we explore the $(q<4,pleqslant q)$ parameter space of finite-sheeted resolvents and derive the associated critical exponents. By definition a value of $q$ is allowed if there is a $p=1$ solution, and we reproduce the long-known result that $q= 2(1+cos{frac{m}{n} pi})$ with $m,n$ coprime. In addition we find that there are two distinct sequences of solutions, one of which contains $p=2$ and $p=q/2$ while the other does not. The boundary condition $p=3$ appears only for $q=3$ which also has a $p=3/2$ boundary condition; we conjecture that this new solution corresponds in the scaling limit to the New boundary condition, discovered on the flat lattice by Affleck et al. We also explore Kramers-Wannier duality for $q=3$ in this context and explicitly construct the known boundary conditions; we show that the mixed boundary condition is dual to a boundary condition on dual graphs that corresponds to Affleck et als identification of the New boundary condition on fixed lattices. On the other hand we find that the mixed boundary condition of the dual, and the corresponding New boundary condition of the original theory are not described by conventional resolvents.
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