Let a building be constructed in the form of a finite undirected connected graph with no other special properties (e.g, not necessarily loopless or planar). Each edge is a corridor, each vertex a circular room. Corridors enter at doors in fixed positions around the room, so that you can tell which corridor is left or right of another with respect to the vertex. You are placed in a random corridor in this building. You are allowed to walk through the building under the restriction that you leave a room at a door left or right of the one which you entered. Rooms and corridors may be identical, so you cannot hope to recognize any room or corridor you have previously visited, nor are you allowed to mark rooms or corridors. Your task is to visit every room in the building. Is there a fixed sequence of left-right choices that will allow you to do this, regardless of the building configuration?