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!