Subclasses or characterizations of modular or pseudo-modular planar graphs
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.