"Sandwich" is purely a descriptor of structure, with no bearing on ingredients. Being "wrapped" and being "sandwiched" are two separate states. Sandwiching is not defined by the state of being surrounded, it is defined by the state of being stuck between two objects. This eilminates tacos, wraps, and all other jokes, and allows ice cream sandwiches, but not open face sandwiches. In the case of open face sandwiches, because "open face" is a structural qualifier, the naming convention of "open face sandwiches" admits that it does not conform to the usual structural definition of sandwich, and so does not invalidate this definition.

Now, this leaves the question of hot dogs and subs, which is where things get dicey. Both are hinge-based structures, but the fundamentals of being stuck between two things is maintained. Shape is irrelevant (see the McRib, undeniably a sandwich). The difference between subs (a sandwich) and hot dogs (not a sandwich) is two-fold: orientation and amount of containment.

A sub is oriented in the same direction as a traditional sandwich, with the slit on the side instead of the bottom. This places the hinge on the opposite side, where a traditional sandwich would only have exposed ingredients. In this case, the hinge has relatively minor impact on the eating experience. If the hinge were to break, the sub is simply a long sandwich. The hinge serves to facilitate the "sandwiching", but is not necessary. On the other hand a hot dog bun is oriented hinge-down. This makes the hinge serve a vital structural role in the hot dog, and this structural role detracts from the "sandwiching". In fact, many hot dogs could be considered to be sitting on the hinge, with the side bread merely providing a convenient means to hold the frankfurter (and a more significant role handling toppings), only truly "sandwiching" the frankfurter where it is held by the hands. Additionally, both the meat and the toppings are much more exposed in a hot dog. As much as half of the frankfurter may be above the "bread line", even more if a bratwurst is used. A sub, although normally exposing more of a surface area of toppings, has minimal excess overhang compared to a hotdog. Therefore, it is reasonable to consider a sub a sandwich, but not a hot dog.