Back Alley Bagel Burger

Sampled a very tasty burger for lunch today - located in one of the city's finest European but not wanky back alley, this little cafe called Jungle Juice serves up not only tasty beverages (squeezed on the spot by expert mixologists), but wonderful smelling soups and sammies.  Although I was interested in the technique of the lovely waitress' squeezing process, the joint is so bustling packed that I feared any distraction might interfere with the procedure and I left my comments to myself.  Back to the burger at hand - an inch and a half patty delicately placed between a piping hot toasted sesame bagel (both clearly prepared right on the spot) with a fine smattering of condiments that were not too juicy with only a four to five drip factor.  A very tasty burger, yet it begs one very important question: Must a burger be placed in between a traditional burger bun?  Does a burger cease to be a burger without a burger bun, thus any burger with an adjective in front of it not a burger? eg bagel burger, focaccia burger etc.  Comments please.