TITLE: Random planar, bipartite graphs
QUESTION [7 upvotes]: I have a need to generate random planar graphs none of which have an odd cycle,
i.e., bipartite 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!

I should have added the only two ideas I had, neither of which I feel is adequate,
even under a loose definition of "random":

Generate a random planar graph, and add a vertex in the middle of
each edge. Then all these newly added vertices have degree $2$.
Select out a random subgraph of the grid graph. Then no vertex has degree
greater than $4$.

REPLY [3 votes]: A further approach starts with a random geometric graph (RGG): Assign a domain $D\subset\mathbb{R}^2$, typically a square.  Place points using a Poisson process and make links between pairs less than a distance $r$ apart.
Then, colour the points randomly with two colours and delete links between points of the same colour to make the graph bipartite.  Then, delete the longest link that intersects another (drawing straight lines between the points) recursively until there are no intersections and the graph is planar.
My interest in RGGs comes from their application to wireless networks (eg arxiv:1401.7188 and references therein); not sure if planar bipartite RGGs have such an application, but in any case it's an interesting question.