If you just go depth, it is well depth first :)

I actually thought this through a bit :-) Neither is it a DFS, unlike in DFS I'm expanding nodes at different depths. E.g., I'm shallow expanding bathroom/kitchen compared to living room. Only when I've exhausted living do I return to kitchen again to do a deeper search. I wonder if this kind of search has any other use. Debugging seems like a similar activity. Go deeper into logs, if nothing turns up then examine metrics, read code, return to logs again etc.,?

Sorry, past midnight here after a busy day at work :-P

Heuristically guided search. :) For some nodes you go wide, for some deep, sometimes you retreat back up a few levels instead of exhausting a subtree (but you might come back to that subtree later), all depending on what the heuristic says is more likely to get you to your goal the quickest. A* is a well-known base algorithm for this, widely used in game programming to efficiently find the shortest path from A to B in the presence of obstacles etc.

A DFS would be you check your living room, you check under the cushions of the sofa, you tear open the cushions and comb through the insides... all before glancing at your bedroom when you didn't quickly see the keys in the living room.