I have a need to generate random planar graphs none of which have an odd cycle,
i.e., *even-cycled* graphs.
I know there is a substantial two-decade literature on random planar graphs, little with which I am familiar.
I know there are several models: at least
those that specify an edge probability, 
those that depend on a random graph process, and
uniform random planar graphs.
I am wondering if those familiar with these and other
models could
suggest one that might be modified to generate even-cycled planar graphs.
I don't have strict distribution requirements—randomness in any one
of several senses would suffice. Thanks for your advice!