[blemepnh] Rooms on a graph

The player is presented with a virtual world of rooms connected by doors.  But the rooms are laid out not in Euclidean space but on an interesting graph: the rooms "weirdly" connect in a way that can be constructed only virtually.

Perhaps the doors are helpfully annotated with "go through this door for the shortest path (distance X) to room Y".

Each room is named, decorated, lit, with music playing to aid memorization.  Despite it being non-Euclidean, will the player be able to form a mental map of the space to be able to get from any room to any other efficiently?

The next challenge is to build this for real.  You only need two rooms, each with (say) 3 doors (for a cubic graph), all leading to the other room.  Only one person at a time can wander through "the complex".  When the person picks which door to go through, the other room has its lighting, sound, etc. automatically changed to become the correct destination room.

It might be helpful to have an adjoining room between the two to ease the implementation, so 6 doors in the middle room: 3 to each side.  Also, to preserve the identity of the doors in each room (i.e., going from one room to another always enters and exits through the same doors in both rooms), the path might cross diagonally through the adjoining room.  Maybe a light in the middle room indicates which door to pass through.  We would especially like to preserve going backwards from one room to the previous uses the same doors as going forwards.

For a stage production, you only need one "room" (the stage), and the connections between rooms are doors leading backstage.  While the protagonist is backstage, the lights (etc.) flip, transforming one room to another room.  There could even be characters present in certain rooms, who enter and exit during the scene changes.  The plot could be the protagonist needs to quickly transport objects among the various rooms.  Or, hunt the wumpus.

The movie "Cube" was filmed this way.

Of particular interest are graphs of a given degree (all rooms have the same number of doors), with small diameter (you can get from any room to any other room fairly quickly).  What is the largest number of rooms?  This is the degree diameter problem.