Subclasses or characterizations of modular or pseudo-modular planar graphs We say that a graph G is modular if for every three vertices x,y,z there exists a vertex w that lies on a shortest path between every two of x, y, z. Pseudo-modular (or "3-Helly") graphs are defined in the following link: http://www.graphclasses.org/classes/gc_203.html. 

I am wondering if there are interesting examples of planar graphs which are modular or pseudo-modular, or an interesting characterization of planar graphs which have these properties.
The median graphs are a subclass of modular graphs with almost the same definition but where the vertex w is always unique. The squaregraphs (plane graphs in which all bounded faces are quadrilaterals and all vertices either have degree 4 or belong to the unbounded face) form a natural and interesting subclass of the planar median graphs. I believe that all planar median graphs can be constructed by gluing together cubes and squaregraphs, but I don't think this result has been published.